{"id":521,"date":"2024-10-18T02:21:45","date_gmt":"2024-10-18T02:21:45","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/mat1260\/?post_type=chapter&#038;p=521"},"modified":"2025-01-08T19:52:59","modified_gmt":"2025-01-08T19:52:59","slug":"6-3-finding-probabilities-for-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/mat1260\/chapter\/6-3-finding-probabilities-for-the-normal-distribution\/","title":{"raw":"6.3: Finding Probabilities for the Normal Distribution","rendered":"6.3: Finding Probabilities for the Normal Distribution"},"content":{"raw":"<div id=\"lobjh\" class=\"\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Learning Objectives<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li id=\"find_probabilities_of_normal_distribution\">Find probabilities associated with the normal distribution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"c0bce3e5ed51469688e562c810645567\" class=\"section\">\r\n<div class=\"sectionContain\">\r\n<h2><span title=\"Quick scroll up\">Finding Probabilities with the Normal Table<\/span><\/h2>\r\n<p id=\"c3efbc3c4a704393a9334e320b282067\">Now that you have learned to assess the relative value of any normal value by standardizing, the next step is to evaluate probabilities. In other contexts, as mentioned before, we will first take the conventional approach of referring to a\u00a0<em>normal table<\/em>, which tells the probability of a normal variable taking a value\u00a0<em>less than<\/em>\u00a0any standardized score z.<\/p>\r\n<p id=\"b708c98ca1e14964a18559ad055aa840\">Click\u00a0<a id=\"normal_table\" class=\"activity_link checkpoint\" href=\"https:\/\/oli.cmu.edu\/jcourse\/webui\/resolver\/link\/resource.do?src=5618ff0c0a0001dc544c93d6919bd7a7&amp;dst=normal_table#\" target=\"new\" rel=\"noopener\">here<\/a> to access the normal table.<\/p>\r\n<p id=\"cd1aefdb2146404fa132f0e525c84b87\">Since normal curves are symmetric about their mean, it follows that the curve of z scores must be symmetric about 0. Since the total area under any normal curve is 1, it follows that the areas on either side of z = 0 are both .5. Also, according to the Standard Deviation Rule, most of the area under the standardized curve falls between z = -3 and z = +3.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"b2d6e88dd3384e8bb5254a9fe23cd2d0\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve. The horizontal axis represents z-scores. The mean&amp;apos;s z-score has been marked as 0, and -3 and 3 have been marked. The area to the right of the mean is .5, and the area to the left of the mean is .5 . Since the area between -3 and 0 is almost all of the area under the bell curve to the left of the mean, that area is approximately .5 . The same goes for the area between 0 and 3.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image133.gif\" alt=\"A normal probability distribution curve. The horizontal axis represents z-scores. The mean&amp;apos;s z-score has been marked as 0, and -3 and 3 have been marked. The area to the right of the mean is .5, and the area to the left of the mean is .5 . Since the area between -3 and 0 is almost all of the area under the bell curve to the left of the mean, that area is approximately .5 . The same goes for the area between 0 and 3.\" \/><\/span><\/span>\r\n<p id=\"f49757ba25f94e04ad4b24206709d0ac\">The normal table outlines the precise behavior of the standard normal random variable Z, the number of standard deviations a normal value x is below or above its mean. The normal table provides probabilities that a standardized normal random variable Z would take a value less than or equal to a particular value z*.<\/p>\r\n<p id=\"c633c69144fa4835808540f0782bf75c\">These particular values are listed in the form *.* in rows along the left margins of the table, specifying the ones and tenths. The columns fine-tune these values to hundredths, allowing us to look up the probability of being below any standardized value z of the form *.**. Here is part of the table.<\/p>\r\n\r\n<table id=\"e58a16ba93304f89a3a1b12d5dd92cb7_bx\" class=\"table labeled\" style=\"height: 638px\">\r\n<tfoot>\r\n<tr style=\"height: 15px\">\r\n<td class=\"captionwrap\" style=\"height: 15px;width: 674.562px\"><\/td>\r\n<\/tr>\r\n<\/tfoot>\r\n<tbody>\r\n<tr style=\"height: 623px\">\r\n<td style=\"height: 623px;width: 674.562px\">\r\n<table id=\"e58a16ba93304f89a3a1b12d5dd92cb7\" class=\"grid aligncenter\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f0616d12dd3c4f8e9403709d2c72b339\">z<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fd2a0282191c412693dbd7c3168eb85f\">.00<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f7464aa4a5644dd1a34b5e6478882407\">.01<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e69c9aba77ab48d88f0040660637c518\">.02<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e5e47d88995c45d985475eee1236a557\">.03<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cde86ba14918479c91ffeda907e52ddd\">.04<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ef7fda98e22b4fdcbb064f09705e4ee3\">.05<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a912385112c540dab6900285c3a0c717\">.06<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dbaeaef192334e20be96a9e2afe03cf9\">.07<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bfaace762b654a39a8f4d119ba04b0f3\">.08<\/p>\r\n<\/th>\r\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a5580ea3858f4509ba8e55ab2763ed01\">.09<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f591e5530ef246f280dd868adfde49ff\">-3.4<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a35f6f7b4af94a469a4092fdb327467d\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cd07ecf462bc46669eb2066ba7268309\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"af6a389a38c44f2fbee4fbebce55c068\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fb21f9e775a04728b3dd45cd08edfd76\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fe61122b138040ecb6cc900951dd39b0\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a9ef394289484aec965433963da3e363\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ae422b9f319a484c8b62c8c4735c3fc5\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f5e52ce3c97e438caf6a107d6b31983e\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fea39e04361c4f63aa1cdbb6cf28b64b\">.0003<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f705198c561b42259f4a899e093a8092\">.0002<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"aba8679618fe465c954f4210e1444250\">-3.3<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fb0ec8f72bb342dfb7a71c17a948c17d\">.0005<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ca9e4a54d9514ee1a4346a5fcbf2ddf2\">.0005<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b9f5f771dd4146c0a609b53ea6b15965\">.0005<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bf4a8367e31b40c5aca9ec07fc7dd44b\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"aeff4661e9894dd899e3beb44031356d\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"becda8b19dc849d484e59a5d1f5eacaf\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cad5247274e146d0be9aa547275033f8\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a68d1410e17945e2983af942399a6a26\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c066682bfe7e4030846aef7afb0f8fab\">.0004<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"eab8649cf4ed4bd1813bd5e7e3d1359d\">.0003<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fda152b15cc8479185e96f89860c97f1\">-3.2<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b6e2f6b3afa649048d06c96e47ed9330\">.0007<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a08e413f8596459284bf668442f99571\">.0007<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cf05af3620de4499a66ec610adc4e502\">.0006<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ce7e84f41a104e078f6f43e9fe4a7f84\">.0006<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c78e7f7298bf4254b7e0ee098ed7917e\">.0006<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fbd28e71581143bb82811a1f1b0a41ff\">.0006<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f07741bc6b9d4e51bb44512c1f31e882\">.0006<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c7968c4428184288957cfb944fe95ef9\">.0005<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dbd07d78144142d0b31522d9c536127f\">.0005<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bdced4075a3747ccadf57c4c0773f13d\">.0005<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"aebaacb425b54efabd23b24f2246ee3b\">-3.1<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c70baef92f5f41c897d194e3d457c363\">.0010<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b3ff056e18be42dca331238c3a4d16e2\">.0009<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"de9b8a23876f49c3b818ef2568798e93\">.0009<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c3dd15889a9142a2a7b794c5f05c96fa\">.0009<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fd651ee503e245eca9d81887eec8ac5e\">.0008<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a6113a8a0ff44cb892d9c5ce89bc518b\">.0008<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e5b0d33012c64690828daca43971ee56\">.0008<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d58365243e3d4a83a4826e487ea31ffe\">.0008<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c8244afc584642e09d08b7f0b9e0a989\">.0007<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f05aa313189b4b4786df5a8498b033b1\">.0007<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b0aaf4b6012a44c6a4f522df15d7e024\">-3.0<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ee557cc15b9f4afeb95e8aeca5ffc74c\">.0013<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c0b2ffe7427946dcaa929be14a3e4f31\">.0013<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ea96e04e72484ff1ac19f10e4bea3ff0\">.0013<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f045fc2a7a774657af4d16472eecb510\">.0012<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f6abb71936c846f196544c4e38b48002\">.0012<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f223f8ba1855497a8f5421f96baa3e55\">.0011<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b09669ec343343079105642ce014b8e9\">.0011<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c533e4b67e2b4926b798a1cb50e180c2\">.0011<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c5cd10aa428a481aa6d12fa26c83d106\">.0010<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cbe825175ad0408bb75bf0c96a95c894\">.0010<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e898ee7032df448bbb5f91bb20c5be47\">-2.9<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fe38878bf39f42469490b3fd1caed54a\">.0019<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b3300895c4b347388705de5cc10c459d\">.0018<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"eb9719fdaf9b4ebba7a780f559db9481\">.0018<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e700adc43c8e45e28e4704d81c7226ed\">.0017<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a186da4a0b764959853351f833531e1d\">.0016<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a945fcb27f994090ac5edbf3ec85a189\">.0016<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"de04f607491b4b80af21c12c5508f0f4\">.0015<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b4f497d982f74d73aabb454e7319795c\">.0015<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cc92cb13dd5e4b0a96a4a741fe1c4b15\">.0014<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c3d426663d334ca5a18f2ec13d471a28\">.0014<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"beb2ca35f8f94779a7c852e1dce273fd\">-2.8<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b3b65a8933a04999aebb443a52630870\">.0026<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ac0c6f30f3334075a938fbc148e1025e\">.0025<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d2ece241b8cc441fb999e1e55b294a67\">.0024<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d822c2d2e0ee4c6cbf28962f317f82ab\">.0023<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f7e699704e2a452fa96b7c6527389c9e\">.0023<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c355b15f1e144555bd309a8472dddd80\">.0022<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d6a10f2135a848eebf561672aa59bda9\">.0021<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ddf98a47f2764da5b231e73909bf25c2\">.0021<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"db897a516d3b4a468f4e6f34c930785a\">.0020<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e1fa40d74dd940d394f0321fcc147a51\">.0019<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fa1b351099954a729c32bf71c61c3c9e\">-2.7<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d0285f4066284420bc6dace3ca5209f0\">.0035<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a78fcf2551c54b749c92a74cb1f30ebd\">.0034<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a6c77230fee84ff68f08fabb8c0906e3\">.0033<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fef5dc4105ff455983538359babdcd57\">.0032<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fd7df3452ae94404907bd4b8aa34da31\">.0031<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ed804ca5ae8d425a9751859423bfbe1a\">.0030<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bfa2692201b04047babed1068861ad97\">.0029<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e3449aba94354e15b528497442d3fd79\">.0028<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fedfc523f96c49e2b1e4e4245ce534e4\">.0027<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d58cc50e56cc439c86d3aa99883d0afb\">.0026<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fde1b2847ca646cda65a6688d1fb19d8\">-2.6<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dc4ffaa59aeb4addb8de5f40dfd3bf67\">.0047<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ad9fb222950148a48b0678725e8f7647\">.0045<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c3e910d78be9472b9baa39ae7f6247b8\">.0044<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f067023b0b4041a88d64bb9c72d8f15e\">.0043<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e19b808b3c5249e69cf3635ae51b8075\">.0041<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d643baf5372d4709a14f697e6c4180f0\">.0040<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a6c42441c7794b88b92eee88fc7ef30e\">.0039<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c2e78e7e980e4654a0413ebeb4dd9c20\">.0038<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e627f45765ea401ca8bb2e86e87b1398\">.0037<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e9b8229499fd4fa597bcb58ae34c0b1f\">.0036<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b241a6a67f4847b9a4d1687e22443466\">-2.5<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e68af617f372446dad303cfbf469d7b9\">.0062<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b4cb32c3bc19461ea25dbaeab669b2da\">.0060<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d57485ae16a04aab9d82be16d53c502c\">.0059<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f324dc8f47c54605a4adba0c30bf8770\">.0057<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d22737759a2845bc857bcf54b1d594f2\">.0055<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a4b20cd06a2d4b61a678548e53b2eeef\">.0054<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b901a234f32f47d5b665fe477fd856b3\">.0052<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b7f923f0927d4c5881a52004c6a3f75f\">.0051<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fa715ac559a14a9a95cc87914800af0f\">.0049<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f20c0fb01a0e4d96a4a669ed9c674945\">.0048<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a0e6d264ccf348b7bd26b97089ee993c\">-2.4<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"abebe257cf4144d883ada1bf652ce30d\">.0082<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ec62f3863dbe460c9305238f9f2fb326\">.0080<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f2103a998f7645a7bf0cf3ba5cb141f0\">.0078<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e3a9b1b12e9b486e86f918e961ccc2c1\">.0075<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a9832c3f1b5140249d1bef403d5cfb95\">.0073<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e863094386a1437a9c3e2b1a0ab23e94\">.0071<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ea926c40b78f48b587cf8636cf0ab4ab\">.0069<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bfae264ee9d1456fb4ad5b4b019b7f64\">.0068<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"cc76142869274beeb710540962bfa88f\">.0066<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dd32a3a919d14435adf27350564ccc0c\">.0064<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d8eaee0d216849e7adbc6ad7a54b9151\">-2.3<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a9f338ca3f6c4370910b3e202ea51799\">.0107<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dadc6de04cf5441e8d0d501a822c92ab\">.0104<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"f41ae27f25084b3eb8b993439f38a080\">.0102<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d20d5d3393934993a96f56d138baa6f8\">.0099<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ad196511b3d349378989dac96fea8180\">.0096<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"fa75c84bdc1a44d58252a08cc901964b\">.0094<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ff6c5fff6bd848cfb74bb7e8a5a6335d\">.0091<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c640dfc8ea364ba48cac5bdd29753a14\">.0089<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a0241e9608c049eeb8a6f41a6809905a\">.0087<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e4ff6529810f457388202793535b7f26\">.0084<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dd847ab53c8945bfa9f4665ec491742c\">-2.2<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e2c05a5d92854d4ba463afa2c6f3bfa7\">.0139<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bc432b22101f4bc0ab33a23c34728991\">.0136<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c5afa164c7034ae1b4b33bca9eb7c04a\">.0132<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a8eed62019184efc9d49a2b6f0ed6da5\">.0129<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e6839fcac7584708b86f8b32aca9c69a\">.0125<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c2783fb411a84fa1bfe1a3f97014d2b1\">.0122<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ca064e2f6dc542178d559a3f4af098b5\">.0119<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ec31004fe2644c5199157cb20450f24a\">.0116<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e344727fed1a48808bc2f8a32a6d7b7a\">.0113<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"bd0b17219120481196298305e9368392\">.0110<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d3a0b3dfbd0c458a99c3f47cdabe377f\">-2.1<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"c2a26b02c8e34495873ecabd9878d361\">.0179<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"e2d1097b67d941d798f71022f4d1b343\">.0174<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"afd65dbbab7c4c1285cecee82ed8ec34\">.0170<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"b5ba3b7a75fb426bad5517bd4da73f85\">.0166<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d85857d78a6b467bb87e73debb5a92a1\">.0162<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"d2cd8f224ffa4d0c9b54b7913c14972d\">.0158<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"dfe8a85da15942ff9ee21045590bb6e6\">.0154<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"a39bc1080d53456286ec309002b1c372\">.0150<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"db7fb263e2c74be8898adfd6397fa6b3\">.0146<\/p>\r\n<\/td>\r\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\r\n<p id=\"ce237d7d21fa4ff18a9a9cdf5e0afdc1\">.0143<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"f77873508e6140619a4187293f2bb775\">By construction, the probability P(Z &lt; z*) equals the area under the z curve to the left of that particular value z*.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"bdd4d6cf243345f39e8db9681ba1f837\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve. The Horizontal axis is in z-score units. On the axis z* is marked, and the area under the curve to the left of z* is shaded. The area is equal to P(Z &amp;lt; z*)\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image135.gif\" alt=\"A normal probability distribution curve. The Horizontal axis is in z-score units. On the axis z* is marked, and the area under the curve to the left of z* is shaded. The area is equal to P(Z &amp;lt; z*)\" \/><\/span><\/span>\r\n<p id=\"d548093b3af248d2a57dedfd173c6509\">A quick sketch is often the key to solving normal problems easily and correctly.<\/p>\r\n\r\n<div class=\"exHead\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Example<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div class=\"exHead\"><\/div>\r\n<div class=\"example clearfix\">\r\n<div>\r\n<p id=\"N10B09\"><em>(a)<\/em>\u00a0What is the probability of a normal random variable taking a value less than 2.8 standard deviations above its mean? According to the table, P(Z &lt; 2.8) = 0.9974 or 99.74%.<\/p>\r\n\r\n<table class=\"grid\" title=\"Standard Normal Probabilities (continued)\">\r\n<thead>\r\n<tr>\r\n<th>z<\/th>\r\n<th>0.00<\/th>\r\n<th>0.01<\/th>\r\n<th>0.02<\/th>\r\n<th>0.03<\/th>\r\n<th>0.04<\/th>\r\n<th>0.05<\/th>\r\n<th>0.06<\/th>\r\n<th>0.07<\/th>\r\n<th>0.08<\/th>\r\n<th>0.09<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2.5<\/td>\r\n<td>0.9938<\/td>\r\n<td>0.9940<\/td>\r\n<td>0.9941<\/td>\r\n<td>0.9943<\/td>\r\n<td>0.9945<\/td>\r\n<td>0.9946<\/td>\r\n<td>0.9948<\/td>\r\n<td>0.9949<\/td>\r\n<td>0.9951<\/td>\r\n<td>0.9952<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>2.6<\/td>\r\n<td>0.9953<\/td>\r\n<td>0.9955<\/td>\r\n<td>0.9956<\/td>\r\n<td>0.9957<\/td>\r\n<td>0.9959<\/td>\r\n<td>0.9960<\/td>\r\n<td>0.9961<\/td>\r\n<td>0.9962<\/td>\r\n<td>0.9963<\/td>\r\n<td>0.9964<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.7<\/td>\r\n<td>0.9965<\/td>\r\n<td>0.9966<\/td>\r\n<td>0.9967<\/td>\r\n<td>0.9968<\/td>\r\n<td>0.9969<\/td>\r\n<td>0.9970<\/td>\r\n<td>0.9971<\/td>\r\n<td>0.9972<\/td>\r\n<td>0.9973<\/td>\r\n<td>0.9974<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<th>2.8<\/th>\r\n<th>0.9974<\/th>\r\n<td>0.9975<\/td>\r\n<td>0.9976<\/td>\r\n<td>0.9977<\/td>\r\n<td>0.9977<\/td>\r\n<td>0.9978<\/td>\r\n<td>0.9979<\/td>\r\n<td>0.9979<\/td>\r\n<td>0.9980<\/td>\r\n<td>0.9981<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.9<\/td>\r\n<td>0.9981<\/td>\r\n<td>0.9982<\/td>\r\n<td>0.9982<\/td>\r\n<td>0.9983<\/td>\r\n<td>0.9984<\/td>\r\n<td>0.9984<\/td>\r\n<td>0.9985<\/td>\r\n<td>0.9985<\/td>\r\n<td>0.9986<\/td>\r\n<td>0.9986<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>3.0<\/td>\r\n<td>0.9987<\/td>\r\n<td>0.9987<\/td>\r\n<td>0.9987<\/td>\r\n<td>0.9988<\/td>\r\n<td>0.9988<\/td>\r\n<td>0.9989<\/td>\r\n<td>0.9989<\/td>\r\n<td>0.9989<\/td>\r\n<td>0.9990<\/td>\r\n<td>0.9990<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_0\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 2.8, and the area to the left of 2.8 under the curve is equal to 0.9974.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image137.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 2.8, and the area to the left of 2.8 under the curve is equal to 0.9974.\" \/><\/span><\/span>\r\n<p id=\"N10C18\"><em>(b)<\/em>\u00a0What is the probability of a normal random variable taking a value lower than 1.47 standard deviations below its mean? P(Z &lt; \u22121.47) = 0.0708, or 7.08%.<\/p>\r\n\r\n<table class=\"grid\" style=\"height: 75px\" title=\"Standard Normal Probabilities\">\r\n<thead>\r\n<tr style=\"height: 15px\">\r\n<th style=\"height: 15px;width: 26.0469px\">z<\/th>\r\n<th style=\"height: 15px;width: 49.0312px\">0.00<\/th>\r\n<th style=\"height: 15px;width: 47.3281px\">0.01<\/th>\r\n<th style=\"height: 15px;width: 47.4844px\">0.02<\/th>\r\n<th style=\"height: 15px;width: 48.6875px\">0.03<\/th>\r\n<th style=\"height: 15px;width: 45.75px\">0.04<\/th>\r\n<th style=\"height: 15px;width: 49px\">0.05<\/th>\r\n<th style=\"height: 15px;width: 48.1094px\">0.06<\/th>\r\n<th style=\"height: 15px;width: 48.7812px\">0.07<\/th>\r\n<th style=\"height: 15px;width: 48.3281px\">0.08<\/th>\r\n<th style=\"height: 15px;width: 48.2031px\">0.09<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 26.5469px\">-1.5<\/td>\r\n<td style=\"height: 15px;width: 50.0312px\">0.0668<\/td>\r\n<td style=\"height: 15px;width: 48.3281px\">0.0655<\/td>\r\n<td style=\"height: 15px;width: 48.4844px\">0.0643<\/td>\r\n<td style=\"height: 15px;width: 49.6875px\">0.0630<\/td>\r\n<td style=\"height: 15px;width: 46.75px\">0.0618<\/td>\r\n<td style=\"height: 15px;width: 50px\">0.0606<\/td>\r\n<td style=\"height: 15px;width: 49.1094px\">0.0594<\/td>\r\n<td style=\"height: 15px;width: 49.7812px\">0.0582<\/td>\r\n<td style=\"height: 15px;width: 49.3281px\">0.0571<\/td>\r\n<td style=\"height: 15px;width: 48.7031px\">0.0559<\/td>\r\n<\/tr>\r\n<tr class=\"e\" style=\"height: 15px\">\r\n<th style=\"height: 15px;width: 26.0469px\">-1.4<\/th>\r\n<td style=\"height: 15px;width: 49.5312px\">0.0808<\/td>\r\n<td style=\"height: 15px;width: 48.3281px\">0.0793<\/td>\r\n<td style=\"height: 15px;width: 48.4844px\">0.0778<\/td>\r\n<td style=\"height: 15px;width: 49.6875px\">0.0764<\/td>\r\n<td style=\"height: 15px;width: 46.75px\">0.0749<\/td>\r\n<td style=\"height: 15px;width: 50px\">0.0735<\/td>\r\n<td style=\"height: 15px;width: 48.6094px\">0.0721<\/td>\r\n<th style=\"height: 15px;width: 48.7812px\">0.0708<\/th>\r\n<td style=\"height: 15px;width: 48.8281px\">0.0694<\/td>\r\n<td style=\"height: 15px;width: 48.7031px\">0.0681<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 26.5469px\">-1.3<\/td>\r\n<td style=\"height: 15px;width: 50.0312px\">0.0968<\/td>\r\n<td style=\"height: 15px;width: 48.3281px\">0.0951<\/td>\r\n<td style=\"height: 15px;width: 48.4844px\">0.0934<\/td>\r\n<td style=\"height: 15px;width: 49.6875px\">0.0918<\/td>\r\n<td style=\"height: 15px;width: 46.75px\">0.0901<\/td>\r\n<td style=\"height: 15px;width: 50px\">0.0885<\/td>\r\n<td style=\"height: 15px;width: 49.1094px\">0.0869<\/td>\r\n<td style=\"height: 15px;width: 49.7812px\">0.0853<\/td>\r\n<td style=\"height: 15px;width: 49.3281px\">0.0838<\/td>\r\n<td style=\"height: 15px;width: 48.7031px\">0.0823<\/td>\r\n<\/tr>\r\n<tr class=\"e\" style=\"height: 15px\">\r\n<td style=\"height: 15px;width: 26.5469px\">-1.2<\/td>\r\n<td style=\"height: 15px;width: 50.0312px\">0.1151<\/td>\r\n<td style=\"height: 15px;width: 48.3281px\">0.1131<\/td>\r\n<td style=\"height: 15px;width: 48.4844px\">0.1112<\/td>\r\n<td style=\"height: 15px;width: 49.6875px\">0.1093<\/td>\r\n<td style=\"height: 15px;width: 46.75px\">0.1075<\/td>\r\n<td style=\"height: 15px;width: 50px\">0.1056<\/td>\r\n<td style=\"height: 15px;width: 49.1094px\">0.1038<\/td>\r\n<td style=\"height: 15px;width: 49.7812px\">0.1020<\/td>\r\n<td style=\"height: 15px;width: 49.3281px\">0.1003<\/td>\r\n<td style=\"height: 15px;width: 48.7031px\">0.0985<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_1\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of \u22121.47, and the area to the left of \u22121.47 under the curve is equal to 0.0708.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image140.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of \u22121.47, and the area to the left of \u22121.47 under the curve is equal to 0.0708.\" \/><\/span><\/span>\r\n<p id=\"N10CDD\"><em>(c)<\/em>\u00a0What is the probability of a normal random variable taking a value\u00a0<em>more<\/em>\u00a0than 0.75 standard deviations above its mean?<\/p>\r\n<p id=\"N10CE5\">The fact that the problem involves the word\u00a0<em class=\"italic\">more<\/em>\u00a0rather than\u00a0<em class=\"italic\">less<\/em>\u00a0should not be overlooked! Our normal table, like most, provides left-tail probabilities, and adjustments must be made for any other type of problem.<\/p>\r\n<p id=\"N10CF0\"><em>Method 1:<\/em>\u00a0By symmetry of the z curve centered on 0, P(Z &gt; + 0.75) = P(Z &lt; \u22120.75) = 0.2266.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_2\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 0.75 and \u22120.75. The area under the curve to the left of \u22120.75 is 0.2266, and since the curve is symmetric, the area under the curve to the right of 0.75 is also 0.2266.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image141.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 0.75 and \u22120.75. The area under the curve to the left of \u22120.75 is 0.2266, and since the curve is symmetric, the area under the curve to the right of 0.75 is also 0.2266.\" \/><\/span><\/span>\r\n<p id=\"N10CFC\"><em>Method 2:<\/em>\u00a0Because the total area under the normal curve is 1,<\/p>\r\n<p id=\"N10D02\">P(Z &gt; + 0.75) = 1 \u2212 P(Z &lt; + 0.75) = 1 \u2212 0.7734 = 0.2266.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_3\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 0.75. The area under the curve to the left of 0.75 is 0.7734, and since the area under the curve is 1, the area to the right of z-score 0.75 is 1 \u2212 0.7734 = 0.2266.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image142.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 0.75. The area under the curve to the left of 0.75 is 0.7734, and since the area under the curve is 1, the area to the right of z-score 0.75 is 1 \u2212 0.7734 = 0.2266.\" \/><\/span><\/span>\r\n<p id=\"N10D0B\"><em>Note:<\/em>\u00a0Most students prefer to use Method 1, which does not require subtracting 4-digit probabilities from 1.<\/p>\r\n<p id=\"N10D10\"><em>(d)<\/em>\u00a0What is the probability of a normal random variable taking a value between 1 standard deviation below and 1 standard deviation above its mean?<\/p>\r\n<p id=\"N10D15\">To find probabilities in between two standard deviations, we must put them in terms of the probabilities below. A sketch is especially helpful here:<\/p>\r\n<p id=\"N10D18\">P(\u22121 &lt; Z &lt; +1) = P(Z &lt; +1) \u2212 P(Z &lt; \u22121) = 0.8413 \u2013 0.1587 = 0.6826.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_4\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 1 and \u22121. The area under the curve to the left of \u22121 is 0.1587, and the area to the left of 1 is 0.8413. To find the area between \u22121 and 1, we subtract: 0.8413 \u2212 0.1587 = 0.6826.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image143.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 1 and \u22121. The area under the curve to the left of \u22121 is 0.1587, and the area to the left of 1 is 0.8413. To find the area between \u22121 and 1, we subtract: 0.8413 \u2212 0.1587 = 0.6826.\" \/><\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Did I get this?<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"134\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\r\n<p id=\"N10B0E\">So far, we have used the normal table to find a probability, given the number (z) of standard deviations below or above the mean. The solution process involved first locating the given z value of the form *.** in the margins, then finding the corresponding probability of the form .**** inside the table as our answer. Now, in Example 2, a probability will be given and we will be asked to find a z value. The solution process involves first locating the given probability of the form .**** inside the table, then finding the corresponding z value of the form *.** as our answer.<\/p>\r\n\r\n<div class=\"example clearfix\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Example<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"N10B13\">(a) The probability is .01 that a standardized normal variable takes a value below what particular value of z?<\/p>\r\n<p id=\"N10B16\">The closest we can come to a probability of .01 inside the table is .0099, in the z = -2.3 row and .03 column: z = -2.33. In other words, the probability is .01 that the value of a normal variable is lower than 2.33 standard deviations below its mean.<\/p>\r\n\r\n<table class=\"grid\" title=\"Standard Normal Probabilities\">\r\n<thead>\r\n<tr>\r\n<th>z<\/th>\r\n<th>.00<\/th>\r\n<th>.01<\/th>\r\n<th>.02<\/th>\r\n<th>.03<\/th>\r\n<th>.04<\/th>\r\n<th>.05<\/th>\r\n<th>.06<\/th>\r\n<th>.07<\/th>\r\n<th>.08<\/th>\r\n<th>.09<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>-2.5<\/td>\r\n<td>.0062<\/td>\r\n<td>.0060<\/td>\r\n<td>.0059<\/td>\r\n<td>.0057<\/td>\r\n<td>.0055<\/td>\r\n<td>.0054<\/td>\r\n<td>.0052<\/td>\r\n<td>.0051<\/td>\r\n<td>.0049<\/td>\r\n<td>.0048<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>-2.4<\/td>\r\n<td>.0082<\/td>\r\n<td>.0080<\/td>\r\n<td>.0078<\/td>\r\n<td>.0075<\/td>\r\n<td>.0073<\/td>\r\n<td>.0071<\/td>\r\n<td>.0069<\/td>\r\n<td>.0068<\/td>\r\n<td>.0066<\/td>\r\n<td>.0064<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-2.3<\/td>\r\n<td>.0107<\/td>\r\n<td>.0104<\/td>\r\n<td>.0102<\/td>\r\n<td>.0099<\/td>\r\n<td>.0096<\/td>\r\n<td>.0094<\/td>\r\n<td>.0091<\/td>\r\n<td>.0089<\/td>\r\n<td>.0087<\/td>\r\n<td>.0084<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>-2.2<\/td>\r\n<td>.0139<\/td>\r\n<td>.0136<\/td>\r\n<td>.0132<\/td>\r\n<td>.0129<\/td>\r\n<td>.0125<\/td>\r\n<td>.0122<\/td>\r\n<td>.0119<\/td>\r\n<td>.0116<\/td>\r\n<td>.0113<\/td>\r\n<td>.0110<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-2.1<\/td>\r\n<td>.0179<\/td>\r\n<td>.0174<\/td>\r\n<td>.0170<\/td>\r\n<td>.0166<\/td>\r\n<td>.0162<\/td>\r\n<td>.0158<\/td>\r\n<td>.0154<\/td>\r\n<td>.0150<\/td>\r\n<td>.0146<\/td>\r\n<td>.0143<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span class=\"imagewrap\"><span class=\"image\"><img class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-scores of -2.33 . The area under the curve to the left of -2.33 is .01.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image145.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-scores of -2.33 . The area under the curve to the left of -2.33 is .01.\" \/><\/span><\/span>\r\n<p id=\"N10BFD\">(b) The probability is .15 that a standardized normal variable takes a value\u00a0<em>above<\/em>\u00a0what particular value of z?<\/p>\r\n<p id=\"N10C03\">Remember that the table only provides probabilities of being\u00a0<em>below<\/em>\u00a0a certain value, not above. Once again, we must rely on one of the properties of the normal curve to make an adjustment.<\/p>\r\n<p id=\"N10C09\"><em>Method 1:<\/em>\u00a0According to the table, .15 (actually .1492) is the probability of being\u00a0<em>below<\/em>\u00a0-1.04. By symmetry, .15 must also be the probability of being\u00a0<em>above<\/em>\u00a0+1.04.<\/p>\r\n\r\n<table class=\"grid\" title=\"Standard Normal Probabilities\">\r\n<thead>\r\n<tr>\r\n<th>z<\/th>\r\n<th>.00<\/th>\r\n<th>.01<\/th>\r\n<th>.02<\/th>\r\n<th>.03<\/th>\r\n<th>.04<\/th>\r\n<th>.05<\/th>\r\n<th>.06<\/th>\r\n<th>.07<\/th>\r\n<th>.08<\/th>\r\n<th>.09<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>-1.2<\/td>\r\n<td>.1151<\/td>\r\n<td>.1131<\/td>\r\n<td>.1112<\/td>\r\n<td>.1093<\/td>\r\n<td>.1075<\/td>\r\n<td>.1056<\/td>\r\n<td>.1038<\/td>\r\n<td>.1020<\/td>\r\n<td>.1003<\/td>\r\n<td>.0985<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>-1.1<\/td>\r\n<td>.1357<\/td>\r\n<td>.1335<\/td>\r\n<td>.1314<\/td>\r\n<td>.1292<\/td>\r\n<td>.1271<\/td>\r\n<td>.1251<\/td>\r\n<td>.1230<\/td>\r\n<td>.1210<\/td>\r\n<td>.1190<\/td>\r\n<td>.1170<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-1.0<\/td>\r\n<td>.1587<\/td>\r\n<td>.1562<\/td>\r\n<td>.1539<\/td>\r\n<td>.1515<\/td>\r\n<td>.1492<\/td>\r\n<td>.1469<\/td>\r\n<td>.1446<\/td>\r\n<td>.1423<\/td>\r\n<td>.1401<\/td>\r\n<td>.1379<\/td>\r\n<\/tr>\r\n<tr class=\"e\">\r\n<td>-0.9<\/td>\r\n<td>.1841<\/td>\r\n<td>.1814<\/td>\r\n<td>.1788<\/td>\r\n<td>.1762<\/td>\r\n<td>.1736<\/td>\r\n<td>.1711<\/td>\r\n<td>.1685<\/td>\r\n<td>.1660<\/td>\r\n<td>.1635<\/td>\r\n<td>.1611<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-0.8<\/td>\r\n<td>.2119<\/td>\r\n<td>.2090<\/td>\r\n<td>.2061<\/td>\r\n<td>.2033<\/td>\r\n<td>.2005<\/td>\r\n<td>.1977<\/td>\r\n<td>.1949<\/td>\r\n<td>.1922<\/td>\r\n<td>.1894<\/td>\r\n<td>.1867<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span class=\"imagewrap\"><span class=\"image\"><img class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of -1.04 and 1.04 . The area under the curve to the left of -1.04 is .15, and since the curve is symmetric, the area under the curve to the right of 1.04 is also .15.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image147.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of -1.04 and 1.04 . The area under the curve to the left of -1.04 is .15, and since the curve is symmetric, the area under the curve to the right of 1.04 is also .15.\" \/><\/span><\/span>\r\n<p id=\"N10CF9\"><em>Method 2:<\/em>\u00a0If .15 is the probability of being above the value we seek, then 1 \u2013 .15 = .85 must be the probability of being below the value we seek. According to the table, .85 (actually .8508) is the probability of being below +1.04.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-score of 1.04. The area under the curve to the right of 1.04 is .15. Knowing this we can calculate the area to the left of 1.04, which is 1-.15 = .85 .\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image148.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-score of 1.04. The area under the curve to the right of 1.04 is .15. Knowing this we can calculate the area to the left of 1.04, which is 1-.15 = .85 .\" \/><\/span><\/span>\r\n<p id=\"N10D05\">In other words, we have found .15 to be the probability that a normal variable takes a value more than 1.04 standard deviations above its mean.<\/p>\r\n<p id=\"N10D08\">(c) The probability is .95 that a normal variable takes a value within how many standard deviations of its mean?<\/p>\r\nA symmetric area of .95 centered at 0 extends to values -z* and +z* such that the remaining (1 \u2013 .95) \/ 2 = .025 is below -z* and also .025 above +z*. The probability is .025 that a standardized normal variable is below -1.96. Thus, the probability is .95 that a normal variable takes a value within 1.96 standard deviations of its mean. Once again, the Standard Deviation Rule is shown to be just roughly accurate, since it states that the probability is .95 that a normal variable takes a value within 2 standard deviations of its mean.\r\n\r\n<span class=\"imagewrap\"><span class=\"image\"><img class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of 1.96 and -1.96 . The area under the curve between these two z-scores is .95 . Since the curve is symmetric, we can calculate the area to the left of -1.96 , which is (1-.95\/2 = .025 . The area to the right of 1.96 is the same.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image149.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of 1.96 and -1.96 . The area under the curve between these two z-scores is .95 . Since the curve is symmetric, we can calculate the area to the left of -1.96 , which is (1-.95\/2 = .025 . The area to the right of 1.96 is the same.\" \/><\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Did I get this?<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"135\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\r\n<p id=\"N10D42\">Our standard normal table, like most, only provides probabilities for z values between -3.49 and +3.49. The following example demonstrates how to handle cases where z exceeds 3.49 in absolute value.<\/p>\r\n\r\n<div class=\"examplewrap\">\r\n<div class=\"example clearfix\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Example<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"N10D47\"><em>(a)<\/em>\u00a0What is the probability of a normal variable being lower than 5.2 standard deviations below its mean?<\/p>\r\n<p id=\"N10D4C\">There is no need to panic about going \u201coff the edge\u201d of the normal table. We already know from the Standard Deviation Rule that the probability is only about (1 \u2013 .997) \/ 2 = .0015 that a normal value would be more than 3 standard deviations away from its mean in one direction or the other. The table provides information for z values as extreme as plus or minus 3.49: the probability is only .0002 that a normal variable would be lower than 3.49 standard deviations below its mean. Any more standard deviations than that, and we generally say the probability is approximately zero.<\/p>\r\n<p id=\"N10D4F\">In this case, we would say the probability of being lower than 5.2 standard deviations below the mean is approximately zero:<\/p>\r\n<p id=\"N10D52\">P(Z &lt; -5.2) = 0 (approx.)<\/p>\r\n<p id=\"N10D55\"><em>(b)<\/em>\u00a0What is the probability of the value of a normal variable being higher than 6 standard deviations below its mean?<\/p>\r\n<p id=\"N10D5A\">Since the probability of being lower than 6 standard deviations below the mean is approximately zero, the probability of being higher than 6 standard deviations below the mean must be approximately 1. P(Z &gt; -6) = 1 (approx.)<\/p>\r\n<p id=\"N10D5D\"><em>(c)<\/em>\u00a0What is the probability of a normal variable being less than 8 standard deviations above the mean? Approximately 1. P(Z &lt; +8) = 1 (approx.)<\/p>\r\n<p id=\"N10D62\"><em>(d)<\/em>\u00a0What is the probability of a normal variable being greater than 3.5 standard deviations above the mean? Approximately 0. P(Z &gt; +3.5) = 0 (approx.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"N10B0F\" class=\"section\">\r\n<div class=\"sectionContain\">\r\n<h2><span title=\"Quick scroll up\">Working with Non-standard Normal Values<\/span><\/h2>\r\n<p id=\"N10B16\">In a much earlier example, we wondered,<\/p>\r\n<p id=\"N10B19\">\u201cHow likely or unlikely is a male foot length of more than 13 inches?\u201d We were unable to solve the problem, because 13 inches didn\u2019t happen to be one of the values featured in the Standard Deviation Rule. Subsequently, we learned how to standardize a normal value (tell how many standard deviations below or above the mean it is) and how to use the normal table to find the probability of falling in an interval a certain number of standard deviations below or above the mean. By combining these two skills, we will now be able to answer questions like the one above.<\/p>\r\n\r\n<div class=\"examplewrap\">\r\n<div class=\"example clearfix\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Example<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<h4>Male Foot Length<\/h4>\r\n<div>\r\n<ol class=\"lower-alpha\">\r\n \t<li>\r\n<p id=\"N10B24\">Male foot lengths have a normal distribution, with\u00a0[latex] \\mu=11, \\sigma=1.5[\/latex]\u00a0inches. What is the probability of a foot length of more than 13 inches?<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_0\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing &quot;foot length X&quot;. The mean is at X=11, and the area which is unknown is the area to the right of X=13 .\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image151.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing &quot;foot length X&quot;. The mean is at X=11, and the area which is unknown is the area to the right of X=13 .\" \/><\/span><\/span>\r\nFirst, we standardize:\u00a0[latex]\\mathcal{z}=\\frac{\\mathcal{x}-\\mu}{\\sigma}=\\frac{13-11}{1.5}=+1.33[\/latex]; the probability that we seek, P(X &gt; 13), is the same as the probability\r\n<p id=\"N10BA7\">P(Z &gt; +1.33) that a normal variable takes a value greater than 1.33 standard deviations above its mean.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_1\" class=\"img-responsive popimg\" title=\"A normal probability distribution curve, with the horizontal axis representing z-scores. The area under the curve to the right of z-score 1.33 is unknown.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image153.gif\" alt=\"A normal probability distribution curve, with the horizontal axis representing z-scores. The area under the curve to the right of z-score 1.33 is unknown.\" \/><\/span><\/span>\r\n<p id=\"N10BB0\">This can be solved with the normal table, after applying the property of symmetry:<\/p>\r\n<p id=\"N10BB3\">P(Z &gt; +1.33) = P(Z &lt; -1.33) = .0918. A male foot length of more than 13 inches is on the long side, but not too unusual: its probability is about 9%.<\/p>\r\n<p id=\"N10BB6\"><em>Comment:<\/em><\/p>\r\n<p id=\"N10BBC\">We can streamline the solution in terms of probability notation. Since the standardized value for 13 is (13 \u2013 11) \/ 1.5 = +1.33, we can write\u00a0[latex]P(X&gt;13)=P(Z&gt;1.33)=P(Z&lt;-1.33)=0.0918[\/latex].<\/p>\r\n<p id=\"N10C2C\">The first equality above holds because we subtracted the mean from a normal variable X and divided by its standard deviation, transforming it to a standardized normal variable that we call \u201cZ.\u201d<\/p>\r\n<p id=\"N10C2F\">The second equality above holds by the symmetry of the standard normal curve around zero.<\/p>\r\n<p id=\"N10C32\">The last equality above was obtained from the normal table.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"N10C36\">What is the probability of a male foot length between 10 and 12 inches?<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_2\" class=\"img-responsive popimg\" title=\"A normal probability distribution, in which the horizontal axis is labeled &quot;foot length x.&quot; The mean is at X = 11. The area under the curve from x = 10 to x = 12 has been shaded. This is the area which we need to find.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image155.gif\" alt=\"A normal probability distribution, in which the horizontal axis is labeled &quot;foot length x.&quot; The mean is at X = 11. The area under the curve from x = 10 to x = 12 has been shaded. This is the area which we need to find.\" \/><\/span><\/span>\r\n<p id=\"N10C3F\">The standardized values of 10 and 12 are, respectively,\u00a0[latex]\\frac{10-11}{1.5}=-0.67[\/latex] and [latex]\\frac{12-11}{1.5}=0.67[\/latex]<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_3\" class=\"img-responsive popimg\" title=\"The same probability distribution curve as above, except that the horizontal axis has been changed to z-score units. At the place where X=10 was, we find the z-score Z=-.67 . And where X=12 was, we find z-score Z=.67 . The area under the curve between these two z-scores has been shaded, marking the unknown area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image158.gif\" alt=\"The same probability distribution curve as above, except that the horizontal axis has been changed to z-score units. At the place where X=10 was, we find the z-score Z=-.67 . And where X=12 was, we find z-score Z=.67 . The area under the curve between these two z-scores has been shaded, marking the unknown area.\" \/><\/span><\/span>\r\n<p id=\"N10CB6\">P(-.67 &lt; Z &lt; +.67) = P(Z &lt; +.67) \u2013 P(Z &lt; -.67) = .7486 \u2013 .2514 = .4972.<\/p>\r\n<p id=\"N10CB9\">Or, if you prefer the streamlined notation,<\/p>\r\n<p id=\"N10CBC\"><span id=\"MathJax-Element-6-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-116\" class=\"mjx-math\"><span id=\"MJXc-Node-117\" class=\"mjx-mrow\"><span id=\"MJXc-Node-118\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-119\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-120\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-121\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-122\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-123\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-124\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-126\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-127\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-128\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-129\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-130\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-131\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-132\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-133\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-134\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-135\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-136\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-137\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-138\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-139\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-140\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-141\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-142\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-143\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-144\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-145\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-146\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-147\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-149\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-150\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-151\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-152\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-153\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-154\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-155\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-156\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-157\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-158\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-161\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-162\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-163\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-164\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-165\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-166\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-167\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-168\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-169\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-170\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-171\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-172\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-173\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-174\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-175\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-176\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-177\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-178\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-179\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-180\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-181\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-182\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-183\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-184\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-185\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-186\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-187\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span>.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"N10D9B\" class=\"section\">\r\n<div class=\"sectionContain\">\r\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\r\n<p id=\"N10DA2\">By solving the above example, we inadvertently discovered the quartiles of a normal distribution! P(Z &lt; -.67) = .2514 tells us that roughly 25%, or one quarter, of a normal variable\u2019s values are less than .67 standard deviations below the mean. P(Z &lt; +.67) = .7486 tells us that roughly 75%, or three quarters, are less than .67 standard deviations above the mean. And of course the median is equal to the mean, since the distribution is symmetric, the median is 0 standard deviations away from the mean.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_4\" class=\"img-responsive popimg\" title=\"A normal probability distribution curve which has a horizontal axis in z-score units. The first quartile is at Z=-.67 . To the left of this, under the curve, is an area of .25 . The second quartile, is at Z=0, the median. To the left of this is an area of .50 under the curve. The third quartile is Z=.67, and to the left is .75 area. To the right of the third quartile is the remaining .25 of area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image160.gif\" alt=\"A normal probability distribution curve which has a horizontal axis in z-score units. The first quartile is at Z=-.67 . To the left of this, under the curve, is an area of .25 . The second quartile, is at Z=0, the median. To the left of this is an area of .50 under the curve. The third quartile is Z=.67, and to the left is .75 area. To the right of the third quartile is the remaining .25 of area.\" \/><\/span><\/span>\r\n<div class=\"examplewrap\">\r\n<div class=\"exHead\"><\/div>\r\n<div class=\"example clearfix\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Example<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<h4>Length of a Human Pregnancy<\/h4>\r\n<div>\r\n<p id=\"N10DAF\">Length (in days) of a randomly chosen human pregnancy is a normal random variable with\u00a0<span id=\"MathJax-Element-7-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-188\" class=\"mjx-math\"><span id=\"MJXc-Node-189\" class=\"mjx-mrow\"><span id=\"MJXc-Node-190\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03bc<\/span><\/span><span id=\"MJXc-Node-191\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-192\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-193\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-195\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-196\" class=\"mjx-mi MJXc-space1\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c3<\/span><\/span><span id=\"MJXc-Node-197\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-198\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-199\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><\/span><\/span><\/p>\r\n\r\n<ol class=\"lower-alpha\">\r\n \t<li>\r\n<p id=\"N10DD8\">Find Q1, the median, and Q3. Q1 = 266 \u2013 .67(16) = 255; median = mean = 266;<\/p>\r\n<p id=\"N10DDB\">Q3 = 266 + .67(16) = 277. Thus, the probability is 1\/4 that a pregnancy will last less than 255 days; 1\/2 that it will last less than 266 days; 3\/4 that it will last less than 277 days.<\/p>\r\n<span class=\"imagewrap\"><span class=\"image\"><img id=\"_i_5\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve on which the horizontal axis is labeled &quot;preg.days x.&quot; The first quartile is marked at 255, with .25 area to the left of it under the curve. The median is marked at 266, with .50 area to the left of it. Lastly, the third quartile is marked at 277, with .75 area to the left of it. In between each quartile is .25 of area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image162.gif\" alt=\"A normal probability distribution curve on which the horizontal axis is labeled &quot;preg.days x.&quot; The first quartile is marked at 255, with .25 area to the left of it under the curve. The median is marked at 266, with .50 area to the left of it. Lastly, the third quartile is marked at 277, with .75 area to the left of it. In between each quartile is .25 of area.\" \/><\/span><\/span><\/li>\r\n \t<li>\r\n<p id=\"N10DE5\">What is the probability that a randomly chosen pregnancy will last less than 246 days? Since (246 \u2013 266) \/ 16 = -1.25, we write<\/p>\r\n<p id=\"N10DE8\"><span id=\"MathJax-Element-8-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-200\" class=\"mjx-math\"><span id=\"MJXc-Node-201\" class=\"mjx-mrow\"><span id=\"MJXc-Node-202\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-203\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-204\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-205\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-206\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-207\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-208\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-209\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-210\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-211\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-212\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-213\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-214\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-215\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-216\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-217\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-218\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-219\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-220\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-221\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-222\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-223\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-224\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-225\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-226\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-227\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><\/span><\/span><\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"N10E3E\">What is the probability that a randomly chosen pregnancy will last longer than 240 days? Since (240 \u2013 266) \/ 16 = -1.63, we write<\/p>\r\n<p id=\"N10E41\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-228\" class=\"mjx-math\"><span id=\"MJXc-Node-229\" class=\"mjx-mrow\"><span id=\"MJXc-Node-230\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-231\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-232\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-233\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-234\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-235\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-236\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-237\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-238\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-239\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-240\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-241\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-242\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-243\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-244\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-245\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-246\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-247\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-248\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-249\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-250\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-251\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-252\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-253\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-254\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-255\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-256\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-257\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-258\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-259\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-260\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-261\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-262\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-263\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-264\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-265\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-266\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/p>\r\n<p id=\"N10EB7\">Since the mean is 266 and the standard deviation is 16, most pregnancies last longer than 240 days.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"N10EBB\">What is the probability that a randomly chosen pregnancy will last longer than 500 days?<\/p>\r\n<p id=\"N10EBE\"><em>Method 1:<\/em>\u00a0Common sense tells us that this would be impossible.<\/p>\r\n<p id=\"N10EC3\"><em>Method 2:<\/em>\u00a0The standardized value of 500 is (500 \u2013 266) \/ 16 = +14.625.\u00a0<span id=\"MathJax-Element-10-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-267\" class=\"mjx-math\"><span id=\"MJXc-Node-268\" class=\"mjx-mrow\"><span id=\"MJXc-Node-269\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-270\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-271\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-272\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-273\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-274\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-275\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-276\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-277\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-278\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-279\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-280\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-281\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-282\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-283\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-284\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-285\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-286\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-287\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-288\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-289\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-290\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"N10F0F\">Suppose a pregnant woman\u2019s husband has scheduled his business trips so that he will be in town between the 235th and 295th days. What is the probability that the birth will take place during that time? The standardized values are (235 \u2013 266) \/ 16) = -1.94 and (295 \u2013 266) \/ 16 = +1.81.<\/p>\r\n<p id=\"N10F12\"><span id=\"MathJax-Element-11-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-291\" class=\"mjx-math\"><span id=\"MJXc-Node-292\" class=\"mjx-mrow\"><span id=\"MJXc-Node-293\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-294\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-295\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-296\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-297\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-298\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-299\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-300\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-301\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-302\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-303\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-304\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-305\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-306\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-307\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-308\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-309\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-310\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-311\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-312\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-313\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-314\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-315\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-316\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-317\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-318\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-319\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-320\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-321\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-322\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-323\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-324\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-325\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-326\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-327\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-328\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-329\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-330\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-331\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-332\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-333\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-334\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-335\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-336\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-337\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-338\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-339\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-340\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-341\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-342\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-343\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-344\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-345\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-346\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-347\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-348\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-349\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-350\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-351\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-352\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-353\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-354\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-355\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-356\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-357\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-358\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-359\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-360\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-361\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-362\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-363\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-364\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><\/span><\/span><\/span><\/p>\r\n<p id=\"N10FF1\">There is close to a 94% chance that the husband will be in town for the birth.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<h3 class=\"textbox__title\">Learn by Doing<\/h3>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccording to the College Board website, the scores on the math part of the SAT (SAT-M) in a certain year had a mean of 507 and standard deviation of 111. Assume that SAT scores follow a normal distribution.\r\n\r\n[h5p id=\"136\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"lobjh\" class=\"\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Learning Objectives<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li id=\"find_probabilities_of_normal_distribution\">Find probabilities associated with the normal distribution.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"c0bce3e5ed51469688e562c810645567\" class=\"section\">\n<div class=\"sectionContain\">\n<h2><span title=\"Quick scroll up\">Finding Probabilities with the Normal Table<\/span><\/h2>\n<p id=\"c3efbc3c4a704393a9334e320b282067\">Now that you have learned to assess the relative value of any normal value by standardizing, the next step is to evaluate probabilities. In other contexts, as mentioned before, we will first take the conventional approach of referring to a\u00a0<em>normal table<\/em>, which tells the probability of a normal variable taking a value\u00a0<em>less than<\/em>\u00a0any standardized score z.<\/p>\n<p id=\"b708c98ca1e14964a18559ad055aa840\">Click\u00a0<a id=\"normal_table\" class=\"activity_link checkpoint\" href=\"https:\/\/oli.cmu.edu\/jcourse\/webui\/resolver\/link\/resource.do?src=5618ff0c0a0001dc544c93d6919bd7a7&amp;dst=normal_table#\" target=\"new\" rel=\"noopener\">here<\/a> to access the normal table.<\/p>\n<p id=\"cd1aefdb2146404fa132f0e525c84b87\">Since normal curves are symmetric about their mean, it follows that the curve of z scores must be symmetric about 0. Since the total area under any normal curve is 1, it follows that the areas on either side of z = 0 are both .5. Also, according to the Standard Deviation Rule, most of the area under the standardized curve falls between z = -3 and z = +3.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"b2d6e88dd3384e8bb5254a9fe23cd2d0\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve. The horizontal axis represents z-scores. The mean&amp;apos;s z-score has been marked as 0, and -3 and 3 have been marked. The area to the right of the mean is .5, and the area to the left of the mean is .5 . Since the area between -3 and 0 is almost all of the area under the bell curve to the left of the mean, that area is approximately .5 . The same goes for the area between 0 and 3.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image133.gif\" alt=\"A normal probability distribution curve. The horizontal axis represents z-scores. The mean&amp;apos;s z-score has been marked as 0, and -3 and 3 have been marked. The area to the right of the mean is .5, and the area to the left of the mean is .5 . Since the area between -3 and 0 is almost all of the area under the bell curve to the left of the mean, that area is approximately .5 . The same goes for the area between 0 and 3.\" \/><\/span><\/span><\/p>\n<p id=\"f49757ba25f94e04ad4b24206709d0ac\">The normal table outlines the precise behavior of the standard normal random variable Z, the number of standard deviations a normal value x is below or above its mean. The normal table provides probabilities that a standardized normal random variable Z would take a value less than or equal to a particular value z*.<\/p>\n<p id=\"c633c69144fa4835808540f0782bf75c\">These particular values are listed in the form *.* in rows along the left margins of the table, specifying the ones and tenths. The columns fine-tune these values to hundredths, allowing us to look up the probability of being below any standardized value z of the form *.**. Here is part of the table.<\/p>\n<table id=\"e58a16ba93304f89a3a1b12d5dd92cb7_bx\" class=\"table labeled\" style=\"height: 638px\">\n<tfoot>\n<tr style=\"height: 15px\">\n<td class=\"captionwrap\" style=\"height: 15px;width: 674.562px\"><\/td>\n<\/tr>\n<\/tfoot>\n<tbody>\n<tr style=\"height: 623px\">\n<td style=\"height: 623px;width: 674.562px\">\n<table id=\"e58a16ba93304f89a3a1b12d5dd92cb7\" class=\"grid aligncenter\">\n<thead>\n<tr>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f0616d12dd3c4f8e9403709d2c72b339\">z<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fd2a0282191c412693dbd7c3168eb85f\">.00<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f7464aa4a5644dd1a34b5e6478882407\">.01<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e69c9aba77ab48d88f0040660637c518\">.02<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e5e47d88995c45d985475eee1236a557\">.03<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cde86ba14918479c91ffeda907e52ddd\">.04<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ef7fda98e22b4fdcbb064f09705e4ee3\">.05<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a912385112c540dab6900285c3a0c717\">.06<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dbaeaef192334e20be96a9e2afe03cf9\">.07<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bfaace762b654a39a8f4d119ba04b0f3\">.08<\/p>\n<\/th>\n<th colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a5580ea3858f4509ba8e55ab2763ed01\">.09<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f591e5530ef246f280dd868adfde49ff\">-3.4<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a35f6f7b4af94a469a4092fdb327467d\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cd07ecf462bc46669eb2066ba7268309\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"af6a389a38c44f2fbee4fbebce55c068\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fb21f9e775a04728b3dd45cd08edfd76\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fe61122b138040ecb6cc900951dd39b0\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a9ef394289484aec965433963da3e363\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ae422b9f319a484c8b62c8c4735c3fc5\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f5e52ce3c97e438caf6a107d6b31983e\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fea39e04361c4f63aa1cdbb6cf28b64b\">.0003<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f705198c561b42259f4a899e093a8092\">.0002<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"aba8679618fe465c954f4210e1444250\">-3.3<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fb0ec8f72bb342dfb7a71c17a948c17d\">.0005<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ca9e4a54d9514ee1a4346a5fcbf2ddf2\">.0005<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b9f5f771dd4146c0a609b53ea6b15965\">.0005<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bf4a8367e31b40c5aca9ec07fc7dd44b\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"aeff4661e9894dd899e3beb44031356d\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"becda8b19dc849d484e59a5d1f5eacaf\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cad5247274e146d0be9aa547275033f8\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a68d1410e17945e2983af942399a6a26\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c066682bfe7e4030846aef7afb0f8fab\">.0004<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"eab8649cf4ed4bd1813bd5e7e3d1359d\">.0003<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fda152b15cc8479185e96f89860c97f1\">-3.2<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b6e2f6b3afa649048d06c96e47ed9330\">.0007<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a08e413f8596459284bf668442f99571\">.0007<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cf05af3620de4499a66ec610adc4e502\">.0006<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ce7e84f41a104e078f6f43e9fe4a7f84\">.0006<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c78e7f7298bf4254b7e0ee098ed7917e\">.0006<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fbd28e71581143bb82811a1f1b0a41ff\">.0006<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f07741bc6b9d4e51bb44512c1f31e882\">.0006<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c7968c4428184288957cfb944fe95ef9\">.0005<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dbd07d78144142d0b31522d9c536127f\">.0005<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bdced4075a3747ccadf57c4c0773f13d\">.0005<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"aebaacb425b54efabd23b24f2246ee3b\">-3.1<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c70baef92f5f41c897d194e3d457c363\">.0010<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b3ff056e18be42dca331238c3a4d16e2\">.0009<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"de9b8a23876f49c3b818ef2568798e93\">.0009<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c3dd15889a9142a2a7b794c5f05c96fa\">.0009<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fd651ee503e245eca9d81887eec8ac5e\">.0008<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a6113a8a0ff44cb892d9c5ce89bc518b\">.0008<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e5b0d33012c64690828daca43971ee56\">.0008<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d58365243e3d4a83a4826e487ea31ffe\">.0008<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c8244afc584642e09d08b7f0b9e0a989\">.0007<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f05aa313189b4b4786df5a8498b033b1\">.0007<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b0aaf4b6012a44c6a4f522df15d7e024\">-3.0<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ee557cc15b9f4afeb95e8aeca5ffc74c\">.0013<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c0b2ffe7427946dcaa929be14a3e4f31\">.0013<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ea96e04e72484ff1ac19f10e4bea3ff0\">.0013<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f045fc2a7a774657af4d16472eecb510\">.0012<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f6abb71936c846f196544c4e38b48002\">.0012<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f223f8ba1855497a8f5421f96baa3e55\">.0011<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b09669ec343343079105642ce014b8e9\">.0011<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c533e4b67e2b4926b798a1cb50e180c2\">.0011<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c5cd10aa428a481aa6d12fa26c83d106\">.0010<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cbe825175ad0408bb75bf0c96a95c894\">.0010<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e898ee7032df448bbb5f91bb20c5be47\">-2.9<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fe38878bf39f42469490b3fd1caed54a\">.0019<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b3300895c4b347388705de5cc10c459d\">.0018<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"eb9719fdaf9b4ebba7a780f559db9481\">.0018<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e700adc43c8e45e28e4704d81c7226ed\">.0017<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a186da4a0b764959853351f833531e1d\">.0016<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a945fcb27f994090ac5edbf3ec85a189\">.0016<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"de04f607491b4b80af21c12c5508f0f4\">.0015<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b4f497d982f74d73aabb454e7319795c\">.0015<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cc92cb13dd5e4b0a96a4a741fe1c4b15\">.0014<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c3d426663d334ca5a18f2ec13d471a28\">.0014<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"beb2ca35f8f94779a7c852e1dce273fd\">-2.8<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b3b65a8933a04999aebb443a52630870\">.0026<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ac0c6f30f3334075a938fbc148e1025e\">.0025<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d2ece241b8cc441fb999e1e55b294a67\">.0024<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d822c2d2e0ee4c6cbf28962f317f82ab\">.0023<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f7e699704e2a452fa96b7c6527389c9e\">.0023<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c355b15f1e144555bd309a8472dddd80\">.0022<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d6a10f2135a848eebf561672aa59bda9\">.0021<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ddf98a47f2764da5b231e73909bf25c2\">.0021<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"db897a516d3b4a468f4e6f34c930785a\">.0020<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e1fa40d74dd940d394f0321fcc147a51\">.0019<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fa1b351099954a729c32bf71c61c3c9e\">-2.7<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d0285f4066284420bc6dace3ca5209f0\">.0035<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a78fcf2551c54b749c92a74cb1f30ebd\">.0034<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a6c77230fee84ff68f08fabb8c0906e3\">.0033<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fef5dc4105ff455983538359babdcd57\">.0032<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fd7df3452ae94404907bd4b8aa34da31\">.0031<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ed804ca5ae8d425a9751859423bfbe1a\">.0030<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bfa2692201b04047babed1068861ad97\">.0029<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e3449aba94354e15b528497442d3fd79\">.0028<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fedfc523f96c49e2b1e4e4245ce534e4\">.0027<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d58cc50e56cc439c86d3aa99883d0afb\">.0026<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fde1b2847ca646cda65a6688d1fb19d8\">-2.6<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dc4ffaa59aeb4addb8de5f40dfd3bf67\">.0047<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ad9fb222950148a48b0678725e8f7647\">.0045<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c3e910d78be9472b9baa39ae7f6247b8\">.0044<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f067023b0b4041a88d64bb9c72d8f15e\">.0043<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e19b808b3c5249e69cf3635ae51b8075\">.0041<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d643baf5372d4709a14f697e6c4180f0\">.0040<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a6c42441c7794b88b92eee88fc7ef30e\">.0039<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c2e78e7e980e4654a0413ebeb4dd9c20\">.0038<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e627f45765ea401ca8bb2e86e87b1398\">.0037<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e9b8229499fd4fa597bcb58ae34c0b1f\">.0036<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b241a6a67f4847b9a4d1687e22443466\">-2.5<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e68af617f372446dad303cfbf469d7b9\">.0062<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b4cb32c3bc19461ea25dbaeab669b2da\">.0060<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d57485ae16a04aab9d82be16d53c502c\">.0059<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f324dc8f47c54605a4adba0c30bf8770\">.0057<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d22737759a2845bc857bcf54b1d594f2\">.0055<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a4b20cd06a2d4b61a678548e53b2eeef\">.0054<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b901a234f32f47d5b665fe477fd856b3\">.0052<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b7f923f0927d4c5881a52004c6a3f75f\">.0051<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fa715ac559a14a9a95cc87914800af0f\">.0049<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f20c0fb01a0e4d96a4a669ed9c674945\">.0048<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a0e6d264ccf348b7bd26b97089ee993c\">-2.4<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"abebe257cf4144d883ada1bf652ce30d\">.0082<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ec62f3863dbe460c9305238f9f2fb326\">.0080<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f2103a998f7645a7bf0cf3ba5cb141f0\">.0078<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e3a9b1b12e9b486e86f918e961ccc2c1\">.0075<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a9832c3f1b5140249d1bef403d5cfb95\">.0073<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e863094386a1437a9c3e2b1a0ab23e94\">.0071<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ea926c40b78f48b587cf8636cf0ab4ab\">.0069<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bfae264ee9d1456fb4ad5b4b019b7f64\">.0068<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"cc76142869274beeb710540962bfa88f\">.0066<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dd32a3a919d14435adf27350564ccc0c\">.0064<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d8eaee0d216849e7adbc6ad7a54b9151\">-2.3<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a9f338ca3f6c4370910b3e202ea51799\">.0107<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dadc6de04cf5441e8d0d501a822c92ab\">.0104<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"f41ae27f25084b3eb8b993439f38a080\">.0102<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d20d5d3393934993a96f56d138baa6f8\">.0099<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ad196511b3d349378989dac96fea8180\">.0096<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"fa75c84bdc1a44d58252a08cc901964b\">.0094<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ff6c5fff6bd848cfb74bb7e8a5a6335d\">.0091<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c640dfc8ea364ba48cac5bdd29753a14\">.0089<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a0241e9608c049eeb8a6f41a6809905a\">.0087<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e4ff6529810f457388202793535b7f26\">.0084<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dd847ab53c8945bfa9f4665ec491742c\">-2.2<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e2c05a5d92854d4ba463afa2c6f3bfa7\">.0139<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bc432b22101f4bc0ab33a23c34728991\">.0136<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c5afa164c7034ae1b4b33bca9eb7c04a\">.0132<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a8eed62019184efc9d49a2b6f0ed6da5\">.0129<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e6839fcac7584708b86f8b32aca9c69a\">.0125<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c2783fb411a84fa1bfe1a3f97014d2b1\">.0122<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ca064e2f6dc542178d559a3f4af098b5\">.0119<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ec31004fe2644c5199157cb20450f24a\">.0116<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e344727fed1a48808bc2f8a32a6d7b7a\">.0113<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"bd0b17219120481196298305e9368392\">.0110<\/p>\n<\/td>\n<\/tr>\n<tr class=\"e\">\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d3a0b3dfbd0c458a99c3f47cdabe377f\">-2.1<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"c2a26b02c8e34495873ecabd9878d361\">.0179<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"e2d1097b67d941d798f71022f4d1b343\">.0174<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"afd65dbbab7c4c1285cecee82ed8ec34\">.0170<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"b5ba3b7a75fb426bad5517bd4da73f85\">.0166<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d85857d78a6b467bb87e73debb5a92a1\">.0162<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"d2cd8f224ffa4d0c9b54b7913c14972d\">.0158<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"dfe8a85da15942ff9ee21045590bb6e6\">.0154<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"a39bc1080d53456286ec309002b1c372\">.0150<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"db7fb263e2c74be8898adfd6397fa6b3\">.0146<\/p>\n<\/td>\n<td colspan=\"1\" rowspan=\"1\" align=\"left\">\n<p id=\"ce237d7d21fa4ff18a9a9cdf5e0afdc1\">.0143<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"f77873508e6140619a4187293f2bb775\">By construction, the probability P(Z &lt; z*) equals the area under the z curve to the left of that particular value z*.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"bdd4d6cf243345f39e8db9681ba1f837\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve. The Horizontal axis is in z-score units. On the axis z* is marked, and the area under the curve to the left of z* is shaded. The area is equal to P(Z &amp;lt; z*)\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image135.gif\" alt=\"A normal probability distribution curve. The Horizontal axis is in z-score units. On the axis z* is marked, and the area under the curve to the left of z* is shaded. The area is equal to P(Z &amp;lt; z*)\" \/><\/span><\/span><\/p>\n<p id=\"d548093b3af248d2a57dedfd173c6509\">A quick sketch is often the key to solving normal problems easily and correctly.<\/p>\n<div class=\"exHead\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Example<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"exHead\"><\/div>\n<div class=\"example clearfix\">\n<div>\n<p id=\"N10B09\"><em>(a)<\/em>\u00a0What is the probability of a normal random variable taking a value less than 2.8 standard deviations above its mean? According to the table, P(Z &lt; 2.8) = 0.9974 or 99.74%.<\/p>\n<table class=\"grid\" title=\"Standard Normal Probabilities (continued)\">\n<thead>\n<tr>\n<th>z<\/th>\n<th>0.00<\/th>\n<th>0.01<\/th>\n<th>0.02<\/th>\n<th>0.03<\/th>\n<th>0.04<\/th>\n<th>0.05<\/th>\n<th>0.06<\/th>\n<th>0.07<\/th>\n<th>0.08<\/th>\n<th>0.09<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2.5<\/td>\n<td>0.9938<\/td>\n<td>0.9940<\/td>\n<td>0.9941<\/td>\n<td>0.9943<\/td>\n<td>0.9945<\/td>\n<td>0.9946<\/td>\n<td>0.9948<\/td>\n<td>0.9949<\/td>\n<td>0.9951<\/td>\n<td>0.9952<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>2.6<\/td>\n<td>0.9953<\/td>\n<td>0.9955<\/td>\n<td>0.9956<\/td>\n<td>0.9957<\/td>\n<td>0.9959<\/td>\n<td>0.9960<\/td>\n<td>0.9961<\/td>\n<td>0.9962<\/td>\n<td>0.9963<\/td>\n<td>0.9964<\/td>\n<\/tr>\n<tr>\n<td>2.7<\/td>\n<td>0.9965<\/td>\n<td>0.9966<\/td>\n<td>0.9967<\/td>\n<td>0.9968<\/td>\n<td>0.9969<\/td>\n<td>0.9970<\/td>\n<td>0.9971<\/td>\n<td>0.9972<\/td>\n<td>0.9973<\/td>\n<td>0.9974<\/td>\n<\/tr>\n<tr class=\"e\">\n<th>2.8<\/th>\n<th>0.9974<\/th>\n<td>0.9975<\/td>\n<td>0.9976<\/td>\n<td>0.9977<\/td>\n<td>0.9977<\/td>\n<td>0.9978<\/td>\n<td>0.9979<\/td>\n<td>0.9979<\/td>\n<td>0.9980<\/td>\n<td>0.9981<\/td>\n<\/tr>\n<tr>\n<td>2.9<\/td>\n<td>0.9981<\/td>\n<td>0.9982<\/td>\n<td>0.9982<\/td>\n<td>0.9983<\/td>\n<td>0.9984<\/td>\n<td>0.9984<\/td>\n<td>0.9985<\/td>\n<td>0.9985<\/td>\n<td>0.9986<\/td>\n<td>0.9986<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>3.0<\/td>\n<td>0.9987<\/td>\n<td>0.9987<\/td>\n<td>0.9987<\/td>\n<td>0.9988<\/td>\n<td>0.9988<\/td>\n<td>0.9989<\/td>\n<td>0.9989<\/td>\n<td>0.9989<\/td>\n<td>0.9990<\/td>\n<td>0.9990<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_0\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 2.8, and the area to the left of 2.8 under the curve is equal to 0.9974.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image137.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 2.8, and the area to the left of 2.8 under the curve is equal to 0.9974.\" \/><\/span><\/span><\/p>\n<p id=\"N10C18\"><em>(b)<\/em>\u00a0What is the probability of a normal random variable taking a value lower than 1.47 standard deviations below its mean? P(Z &lt; \u22121.47) = 0.0708, or 7.08%.<\/p>\n<table class=\"grid\" style=\"height: 75px\" title=\"Standard Normal Probabilities\">\n<thead>\n<tr style=\"height: 15px\">\n<th style=\"height: 15px;width: 26.0469px\">z<\/th>\n<th style=\"height: 15px;width: 49.0312px\">0.00<\/th>\n<th style=\"height: 15px;width: 47.3281px\">0.01<\/th>\n<th style=\"height: 15px;width: 47.4844px\">0.02<\/th>\n<th style=\"height: 15px;width: 48.6875px\">0.03<\/th>\n<th style=\"height: 15px;width: 45.75px\">0.04<\/th>\n<th style=\"height: 15px;width: 49px\">0.05<\/th>\n<th style=\"height: 15px;width: 48.1094px\">0.06<\/th>\n<th style=\"height: 15px;width: 48.7812px\">0.07<\/th>\n<th style=\"height: 15px;width: 48.3281px\">0.08<\/th>\n<th style=\"height: 15px;width: 48.2031px\">0.09<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 26.5469px\">-1.5<\/td>\n<td style=\"height: 15px;width: 50.0312px\">0.0668<\/td>\n<td style=\"height: 15px;width: 48.3281px\">0.0655<\/td>\n<td style=\"height: 15px;width: 48.4844px\">0.0643<\/td>\n<td style=\"height: 15px;width: 49.6875px\">0.0630<\/td>\n<td style=\"height: 15px;width: 46.75px\">0.0618<\/td>\n<td style=\"height: 15px;width: 50px\">0.0606<\/td>\n<td style=\"height: 15px;width: 49.1094px\">0.0594<\/td>\n<td style=\"height: 15px;width: 49.7812px\">0.0582<\/td>\n<td style=\"height: 15px;width: 49.3281px\">0.0571<\/td>\n<td style=\"height: 15px;width: 48.7031px\">0.0559<\/td>\n<\/tr>\n<tr class=\"e\" style=\"height: 15px\">\n<th style=\"height: 15px;width: 26.0469px\">-1.4<\/th>\n<td style=\"height: 15px;width: 49.5312px\">0.0808<\/td>\n<td style=\"height: 15px;width: 48.3281px\">0.0793<\/td>\n<td style=\"height: 15px;width: 48.4844px\">0.0778<\/td>\n<td style=\"height: 15px;width: 49.6875px\">0.0764<\/td>\n<td style=\"height: 15px;width: 46.75px\">0.0749<\/td>\n<td style=\"height: 15px;width: 50px\">0.0735<\/td>\n<td style=\"height: 15px;width: 48.6094px\">0.0721<\/td>\n<th style=\"height: 15px;width: 48.7812px\">0.0708<\/th>\n<td style=\"height: 15px;width: 48.8281px\">0.0694<\/td>\n<td style=\"height: 15px;width: 48.7031px\">0.0681<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px;width: 26.5469px\">-1.3<\/td>\n<td style=\"height: 15px;width: 50.0312px\">0.0968<\/td>\n<td style=\"height: 15px;width: 48.3281px\">0.0951<\/td>\n<td style=\"height: 15px;width: 48.4844px\">0.0934<\/td>\n<td style=\"height: 15px;width: 49.6875px\">0.0918<\/td>\n<td style=\"height: 15px;width: 46.75px\">0.0901<\/td>\n<td style=\"height: 15px;width: 50px\">0.0885<\/td>\n<td style=\"height: 15px;width: 49.1094px\">0.0869<\/td>\n<td style=\"height: 15px;width: 49.7812px\">0.0853<\/td>\n<td style=\"height: 15px;width: 49.3281px\">0.0838<\/td>\n<td style=\"height: 15px;width: 48.7031px\">0.0823<\/td>\n<\/tr>\n<tr class=\"e\" style=\"height: 15px\">\n<td style=\"height: 15px;width: 26.5469px\">-1.2<\/td>\n<td style=\"height: 15px;width: 50.0312px\">0.1151<\/td>\n<td style=\"height: 15px;width: 48.3281px\">0.1131<\/td>\n<td style=\"height: 15px;width: 48.4844px\">0.1112<\/td>\n<td style=\"height: 15px;width: 49.6875px\">0.1093<\/td>\n<td style=\"height: 15px;width: 46.75px\">0.1075<\/td>\n<td style=\"height: 15px;width: 50px\">0.1056<\/td>\n<td style=\"height: 15px;width: 49.1094px\">0.1038<\/td>\n<td style=\"height: 15px;width: 49.7812px\">0.1020<\/td>\n<td style=\"height: 15px;width: 49.3281px\">0.1003<\/td>\n<td style=\"height: 15px;width: 48.7031px\">0.0985<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_1\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of \u22121.47, and the area to the left of \u22121.47 under the curve is equal to 0.0708.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image140.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of \u22121.47, and the area to the left of \u22121.47 under the curve is equal to 0.0708.\" \/><\/span><\/span><\/p>\n<p id=\"N10CDD\"><em>(c)<\/em>\u00a0What is the probability of a normal random variable taking a value\u00a0<em>more<\/em>\u00a0than 0.75 standard deviations above its mean?<\/p>\n<p id=\"N10CE5\">The fact that the problem involves the word\u00a0<em class=\"italic\">more<\/em>\u00a0rather than\u00a0<em class=\"italic\">less<\/em>\u00a0should not be overlooked! Our normal table, like most, provides left-tail probabilities, and adjustments must be made for any other type of problem.<\/p>\n<p id=\"N10CF0\"><em>Method 1:<\/em>\u00a0By symmetry of the z curve centered on 0, P(Z &gt; + 0.75) = P(Z &lt; \u22120.75) = 0.2266.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_2\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 0.75 and \u22120.75. The area under the curve to the left of \u22120.75 is 0.2266, and since the curve is symmetric, the area under the curve to the right of 0.75 is also 0.2266.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image141.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 0.75 and \u22120.75. The area under the curve to the left of \u22120.75 is 0.2266, and since the curve is symmetric, the area under the curve to the right of 0.75 is also 0.2266.\" \/><\/span><\/span><\/p>\n<p id=\"N10CFC\"><em>Method 2:<\/em>\u00a0Because the total area under the normal curve is 1,<\/p>\n<p id=\"N10D02\">P(Z &gt; + 0.75) = 1 \u2212 P(Z &lt; + 0.75) = 1 \u2212 0.7734 = 0.2266.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_3\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 0.75. The area under the curve to the left of 0.75 is 0.7734, and since the area under the curve is 1, the area to the right of z-score 0.75 is 1 \u2212 0.7734 = 0.2266.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image142.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis is a z-score of 0.75. The area under the curve to the left of 0.75 is 0.7734, and since the area under the curve is 1, the area to the right of z-score 0.75 is 1 \u2212 0.7734 = 0.2266.\" \/><\/span><\/span><\/p>\n<p id=\"N10D0B\"><em>Note:<\/em>\u00a0Most students prefer to use Method 1, which does not require subtracting 4-digit probabilities from 1.<\/p>\n<p id=\"N10D10\"><em>(d)<\/em>\u00a0What is the probability of a normal random variable taking a value between 1 standard deviation below and 1 standard deviation above its mean?<\/p>\n<p id=\"N10D15\">To find probabilities in between two standard deviations, we must put them in terms of the probabilities below. A sketch is especially helpful here:<\/p>\n<p id=\"N10D18\">P(\u22121 &lt; Z &lt; +1) = P(Z &lt; +1) \u2212 P(Z &lt; \u22121) = 0.8413 \u2013 0.1587 = 0.6826.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_4\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 1 and \u22121. The area under the curve to the left of \u22121 is 0.1587, and the area to the left of 1 is 0.8413. To find the area between \u22121 and 1, we subtract: 0.8413 \u2212 0.1587 = 0.6826.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image143.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores. Marked on the axis are z-scores of 1 and \u22121. The area under the curve to the left of \u22121 is 0.1587, and the area to the left of 1 is 0.8413. To find the area between \u22121 and 1, we subtract: 0.8413 \u2212 0.1587 = 0.6826.\" \/><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Did I get this?<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-134\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-134\" class=\"h5p-iframe\" data-content-id=\"134\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"6.3 Did I get this?\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\n<p id=\"N10B0E\">So far, we have used the normal table to find a probability, given the number (z) of standard deviations below or above the mean. The solution process involved first locating the given z value of the form *.** in the margins, then finding the corresponding probability of the form .**** inside the table as our answer. Now, in Example 2, a probability will be given and we will be asked to find a z value. The solution process involves first locating the given probability of the form .**** inside the table, then finding the corresponding z value of the form *.** as our answer.<\/p>\n<div class=\"example clearfix\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Example<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"N10B13\">(a) The probability is .01 that a standardized normal variable takes a value below what particular value of z?<\/p>\n<p id=\"N10B16\">The closest we can come to a probability of .01 inside the table is .0099, in the z = -2.3 row and .03 column: z = -2.33. In other words, the probability is .01 that the value of a normal variable is lower than 2.33 standard deviations below its mean.<\/p>\n<table class=\"grid\" title=\"Standard Normal Probabilities\">\n<thead>\n<tr>\n<th>z<\/th>\n<th>.00<\/th>\n<th>.01<\/th>\n<th>.02<\/th>\n<th>.03<\/th>\n<th>.04<\/th>\n<th>.05<\/th>\n<th>.06<\/th>\n<th>.07<\/th>\n<th>.08<\/th>\n<th>.09<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>-2.5<\/td>\n<td>.0062<\/td>\n<td>.0060<\/td>\n<td>.0059<\/td>\n<td>.0057<\/td>\n<td>.0055<\/td>\n<td>.0054<\/td>\n<td>.0052<\/td>\n<td>.0051<\/td>\n<td>.0049<\/td>\n<td>.0048<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>-2.4<\/td>\n<td>.0082<\/td>\n<td>.0080<\/td>\n<td>.0078<\/td>\n<td>.0075<\/td>\n<td>.0073<\/td>\n<td>.0071<\/td>\n<td>.0069<\/td>\n<td>.0068<\/td>\n<td>.0066<\/td>\n<td>.0064<\/td>\n<\/tr>\n<tr>\n<td>-2.3<\/td>\n<td>.0107<\/td>\n<td>.0104<\/td>\n<td>.0102<\/td>\n<td>.0099<\/td>\n<td>.0096<\/td>\n<td>.0094<\/td>\n<td>.0091<\/td>\n<td>.0089<\/td>\n<td>.0087<\/td>\n<td>.0084<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>-2.2<\/td>\n<td>.0139<\/td>\n<td>.0136<\/td>\n<td>.0132<\/td>\n<td>.0129<\/td>\n<td>.0125<\/td>\n<td>.0122<\/td>\n<td>.0119<\/td>\n<td>.0116<\/td>\n<td>.0113<\/td>\n<td>.0110<\/td>\n<\/tr>\n<tr>\n<td>-2.1<\/td>\n<td>.0179<\/td>\n<td>.0174<\/td>\n<td>.0170<\/td>\n<td>.0166<\/td>\n<td>.0162<\/td>\n<td>.0158<\/td>\n<td>.0154<\/td>\n<td>.0150<\/td>\n<td>.0146<\/td>\n<td>.0143<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-scores of -2.33 . The area under the curve to the left of -2.33 is .01.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image145.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-scores of -2.33 . The area under the curve to the left of -2.33 is .01.\" \/><\/span><\/span><\/p>\n<p id=\"N10BFD\">(b) The probability is .15 that a standardized normal variable takes a value\u00a0<em>above<\/em>\u00a0what particular value of z?<\/p>\n<p id=\"N10C03\">Remember that the table only provides probabilities of being\u00a0<em>below<\/em>\u00a0a certain value, not above. Once again, we must rely on one of the properties of the normal curve to make an adjustment.<\/p>\n<p id=\"N10C09\"><em>Method 1:<\/em>\u00a0According to the table, .15 (actually .1492) is the probability of being\u00a0<em>below<\/em>\u00a0-1.04. By symmetry, .15 must also be the probability of being\u00a0<em>above<\/em>\u00a0+1.04.<\/p>\n<table class=\"grid\" title=\"Standard Normal Probabilities\">\n<thead>\n<tr>\n<th>z<\/th>\n<th>.00<\/th>\n<th>.01<\/th>\n<th>.02<\/th>\n<th>.03<\/th>\n<th>.04<\/th>\n<th>.05<\/th>\n<th>.06<\/th>\n<th>.07<\/th>\n<th>.08<\/th>\n<th>.09<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>-1.2<\/td>\n<td>.1151<\/td>\n<td>.1131<\/td>\n<td>.1112<\/td>\n<td>.1093<\/td>\n<td>.1075<\/td>\n<td>.1056<\/td>\n<td>.1038<\/td>\n<td>.1020<\/td>\n<td>.1003<\/td>\n<td>.0985<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>-1.1<\/td>\n<td>.1357<\/td>\n<td>.1335<\/td>\n<td>.1314<\/td>\n<td>.1292<\/td>\n<td>.1271<\/td>\n<td>.1251<\/td>\n<td>.1230<\/td>\n<td>.1210<\/td>\n<td>.1190<\/td>\n<td>.1170<\/td>\n<\/tr>\n<tr>\n<td>-1.0<\/td>\n<td>.1587<\/td>\n<td>.1562<\/td>\n<td>.1539<\/td>\n<td>.1515<\/td>\n<td>.1492<\/td>\n<td>.1469<\/td>\n<td>.1446<\/td>\n<td>.1423<\/td>\n<td>.1401<\/td>\n<td>.1379<\/td>\n<\/tr>\n<tr class=\"e\">\n<td>-0.9<\/td>\n<td>.1841<\/td>\n<td>.1814<\/td>\n<td>.1788<\/td>\n<td>.1762<\/td>\n<td>.1736<\/td>\n<td>.1711<\/td>\n<td>.1685<\/td>\n<td>.1660<\/td>\n<td>.1635<\/td>\n<td>.1611<\/td>\n<\/tr>\n<tr>\n<td>-0.8<\/td>\n<td>.2119<\/td>\n<td>.2090<\/td>\n<td>.2061<\/td>\n<td>.2033<\/td>\n<td>.2005<\/td>\n<td>.1977<\/td>\n<td>.1949<\/td>\n<td>.1922<\/td>\n<td>.1894<\/td>\n<td>.1867<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of -1.04 and 1.04 . The area under the curve to the left of -1.04 is .15, and since the curve is symmetric, the area under the curve to the right of 1.04 is also .15.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image147.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of -1.04 and 1.04 . The area under the curve to the left of -1.04 is .15, and since the curve is symmetric, the area under the curve to the right of 1.04 is also .15.\" \/><\/span><\/span><\/p>\n<p id=\"N10CF9\"><em>Method 2:<\/em>\u00a0If .15 is the probability of being above the value we seek, then 1 \u2013 .15 = .85 must be the probability of being below the value we seek. According to the table, .85 (actually .8508) is the probability of being below +1.04.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-score of 1.04. The area under the curve to the right of 1.04 is .15. Knowing this we can calculate the area to the left of 1.04, which is 1-.15 = .85 .\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image148.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis is a z-score of 1.04. The area under the curve to the right of 1.04 is .15. Knowing this we can calculate the area to the left of 1.04, which is 1-.15 = .85 .\" \/><\/span><\/span><\/p>\n<p id=\"N10D05\">In other words, we have found .15 to be the probability that a normal variable takes a value more than 1.04 standard deviations above its mean.<\/p>\n<p id=\"N10D08\">(c) The probability is .95 that a normal variable takes a value within how many standard deviations of its mean?<\/p>\n<p>A symmetric area of .95 centered at 0 extends to values -z* and +z* such that the remaining (1 \u2013 .95) \/ 2 = .025 is below -z* and also .025 above +z*. The probability is .025 that a standardized normal variable is below -1.96. Thus, the probability is .95 that a normal variable takes a value within 1.96 standard deviations of its mean. Once again, the Standard Deviation Rule is shown to be just roughly accurate, since it states that the probability is .95 that a normal variable takes a value within 2 standard deviations of its mean.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of 1.96 and -1.96 . The area under the curve between these two z-scores is .95 . Since the curve is symmetric, we can calculate the area to the left of -1.96 , which is (1-.95\/2 = .025 . The area to the right of 1.96 is the same.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image149.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing z-scores . Marked on the axis are z-scores of 1.96 and -1.96 . The area under the curve between these two z-scores is .95 . Since the curve is symmetric, we can calculate the area to the left of -1.96 , which is (1-.95\/2 = .025 . The area to the right of 1.96 is the same.\" \/><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Did I get this?<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-135\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-135\" class=\"h5p-iframe\" data-content-id=\"135\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"6.3 Did I get this? 2\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\n<p id=\"N10D42\">Our standard normal table, like most, only provides probabilities for z values between -3.49 and +3.49. The following example demonstrates how to handle cases where z exceeds 3.49 in absolute value.<\/p>\n<div class=\"examplewrap\">\n<div class=\"example clearfix\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Example<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"N10D47\"><em>(a)<\/em>\u00a0What is the probability of a normal variable being lower than 5.2 standard deviations below its mean?<\/p>\n<p id=\"N10D4C\">There is no need to panic about going \u201coff the edge\u201d of the normal table. We already know from the Standard Deviation Rule that the probability is only about (1 \u2013 .997) \/ 2 = .0015 that a normal value would be more than 3 standard deviations away from its mean in one direction or the other. The table provides information for z values as extreme as plus or minus 3.49: the probability is only .0002 that a normal variable would be lower than 3.49 standard deviations below its mean. Any more standard deviations than that, and we generally say the probability is approximately zero.<\/p>\n<p id=\"N10D4F\">In this case, we would say the probability of being lower than 5.2 standard deviations below the mean is approximately zero:<\/p>\n<p id=\"N10D52\">P(Z &lt; -5.2) = 0 (approx.)<\/p>\n<p id=\"N10D55\"><em>(b)<\/em>\u00a0What is the probability of the value of a normal variable being higher than 6 standard deviations below its mean?<\/p>\n<p id=\"N10D5A\">Since the probability of being lower than 6 standard deviations below the mean is approximately zero, the probability of being higher than 6 standard deviations below the mean must be approximately 1. P(Z &gt; -6) = 1 (approx.)<\/p>\n<p id=\"N10D5D\"><em>(c)<\/em>\u00a0What is the probability of a normal variable being less than 8 standard deviations above the mean? Approximately 1. P(Z &lt; +8) = 1 (approx.)<\/p>\n<p id=\"N10D62\"><em>(d)<\/em>\u00a0What is the probability of a normal variable being greater than 3.5 standard deviations above the mean? Approximately 0. P(Z &gt; +3.5) = 0 (approx.)<\/p>\n<\/div>\n<\/div>\n<div id=\"N10B0F\" class=\"section\">\n<div class=\"sectionContain\">\n<h2><span title=\"Quick scroll up\">Working with Non-standard Normal Values<\/span><\/h2>\n<p>In a much earlier example, we wondered,<\/p>\n<p id=\"N10B19\">\u201cHow likely or unlikely is a male foot length of more than 13 inches?\u201d We were unable to solve the problem, because 13 inches didn\u2019t happen to be one of the values featured in the Standard Deviation Rule. Subsequently, we learned how to standardize a normal value (tell how many standard deviations below or above the mean it is) and how to use the normal table to find the probability of falling in an interval a certain number of standard deviations below or above the mean. By combining these two skills, we will now be able to answer questions like the one above.<\/p>\n<div class=\"examplewrap\">\n<div class=\"example clearfix\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Example<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<h4>Male Foot Length<\/h4>\n<div>\n<ol class=\"lower-alpha\">\n<li>\n<p id=\"N10B24\">Male foot lengths have a normal distribution, with\u00a0[latex]\\mu=11, \\sigma=1.5[\/latex]\u00a0inches. What is the probability of a foot length of more than 13 inches?<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve with its horizontal axis representing &quot;foot length X&quot;. The mean is at X=11, and the area which is unknown is the area to the right of X=13 .\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image151.gif\" alt=\"A normal probability distribution curve with its horizontal axis representing &quot;foot length X&quot;. The mean is at X=11, and the area which is unknown is the area to the right of X=13 .\" \/><\/span><\/span><br \/>\nFirst, we standardize:\u00a0[latex]\\mathcal{z}=\\frac{\\mathcal{x}-\\mu}{\\sigma}=\\frac{13-11}{1.5}=+1.33[\/latex]; the probability that we seek, P(X &gt; 13), is the same as the probability<\/p>\n<p id=\"N10BA7\">P(Z &gt; +1.33) that a normal variable takes a value greater than 1.33 standard deviations above its mean.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg\" title=\"A normal probability distribution curve, with the horizontal axis representing z-scores. The area under the curve to the right of z-score 1.33 is unknown.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image153.gif\" alt=\"A normal probability distribution curve, with the horizontal axis representing z-scores. The area under the curve to the right of z-score 1.33 is unknown.\" \/><\/span><\/span><\/p>\n<p id=\"N10BB0\">This can be solved with the normal table, after applying the property of symmetry:<\/p>\n<p id=\"N10BB3\">P(Z &gt; +1.33) = P(Z &lt; -1.33) = .0918. A male foot length of more than 13 inches is on the long side, but not too unusual: its probability is about 9%.<\/p>\n<p id=\"N10BB6\"><em>Comment:<\/em><\/p>\n<p id=\"N10BBC\">We can streamline the solution in terms of probability notation. Since the standardized value for 13 is (13 \u2013 11) \/ 1.5 = +1.33, we can write\u00a0[latex]P(X>13)=P(Z>1.33)=P(Z<-1.33)=0.0918[\/latex].<\/p>\n<p id=\"N10C2C\">The first equality above holds because we subtracted the mean from a normal variable X and divided by its standard deviation, transforming it to a standardized normal variable that we call \u201cZ.\u201d<\/p>\n<p id=\"N10C2F\">The second equality above holds by the symmetry of the standard normal curve around zero.<\/p>\n<p id=\"N10C32\">The last equality above was obtained from the normal table.<\/p>\n<\/li>\n<li>\n<p id=\"N10C36\">What is the probability of a male foot length between 10 and 12 inches?<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg\" title=\"A normal probability distribution, in which the horizontal axis is labeled &quot;foot length x.&quot; The mean is at X = 11. The area under the curve from x = 10 to x = 12 has been shaded. This is the area which we need to find.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image155.gif\" alt=\"A normal probability distribution, in which the horizontal axis is labeled &quot;foot length x.&quot; The mean is at X = 11. The area under the curve from x = 10 to x = 12 has been shaded. This is the area which we need to find.\" \/><\/span><\/span><\/p>\n<p id=\"N10C3F\">The standardized values of 10 and 12 are, respectively,\u00a0[latex]\\frac{10-11}{1.5}=-0.67[\/latex] and [latex]\\frac{12-11}{1.5}=0.67[\/latex]<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg\" title=\"The same probability distribution curve as above, except that the horizontal axis has been changed to z-score units. At the place where X=10 was, we find the z-score Z=-.67 . And where X=12 was, we find z-score Z=.67 . The area under the curve between these two z-scores has been shaded, marking the unknown area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image158.gif\" alt=\"The same probability distribution curve as above, except that the horizontal axis has been changed to z-score units. At the place where X=10 was, we find the z-score Z=-.67 . And where X=12 was, we find z-score Z=.67 . The area under the curve between these two z-scores has been shaded, marking the unknown area.\" \/><\/span><\/span><\/p>\n<p id=\"N10CB6\">P(-.67 &lt; Z &lt; +.67) = P(Z &lt; +.67) \u2013 P(Z &lt; -.67) = .7486 \u2013 .2514 = .4972.<\/p>\n<p id=\"N10CB9\">Or, if you prefer the streamlined notation,<\/p>\n<p id=\"N10CBC\"><span id=\"MathJax-Element-6-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-116\" class=\"mjx-math\"><span id=\"MJXc-Node-117\" class=\"mjx-mrow\"><span id=\"MJXc-Node-118\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-119\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-120\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-121\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-122\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-123\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-124\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-126\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-127\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-128\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-129\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-130\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-131\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-132\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-133\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-134\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-135\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-136\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-137\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-138\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-139\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-140\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-141\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-142\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-143\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-144\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-145\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-146\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-147\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-149\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-150\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-151\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-152\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-153\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-154\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-155\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-156\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-157\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-158\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-161\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-162\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-163\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-164\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-165\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-166\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-167\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-168\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-169\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-170\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-171\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-172\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-173\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-174\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-175\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-176\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-177\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-178\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-179\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-180\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-181\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-182\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-183\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-184\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-185\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-186\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><span id=\"MJXc-Node-187\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"N10D9B\" class=\"section\">\n<div class=\"sectionContain\">\n<h2><span title=\"Quick scroll up\">Comment<\/span><\/h2>\n<p id=\"N10DA2\">By solving the above example, we inadvertently discovered the quartiles of a normal distribution! P(Z &lt; -.67) = .2514 tells us that roughly 25%, or one quarter, of a normal variable\u2019s values are less than .67 standard deviations below the mean. P(Z &lt; +.67) = .7486 tells us that roughly 75%, or three quarters, are less than .67 standard deviations above the mean. And of course the median is equal to the mean, since the distribution is symmetric, the median is 0 standard deviations away from the mean.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" class=\"img-responsive popimg\" title=\"A normal probability distribution curve which has a horizontal axis in z-score units. The first quartile is at Z=-.67 . To the left of this, under the curve, is an area of .25 . The second quartile, is at Z=0, the median. To the left of this is an area of .50 under the curve. The third quartile is Z=.67, and to the left is .75 area. To the right of the third quartile is the remaining .25 of area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image160.gif\" alt=\"A normal probability distribution curve which has a horizontal axis in z-score units. The first quartile is at Z=-.67 . To the left of this, under the curve, is an area of .25 . The second quartile, is at Z=0, the median. To the left of this is an area of .50 under the curve. The third quartile is Z=.67, and to the left is .75 area. To the right of the third quartile is the remaining .25 of area.\" \/><\/span><\/span><\/p>\n<div class=\"examplewrap\">\n<div class=\"exHead\"><\/div>\n<div class=\"example clearfix\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Example<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<h4>Length of a Human Pregnancy<\/h4>\n<div>\n<p id=\"N10DAF\">Length (in days) of a randomly chosen human pregnancy is a normal random variable with\u00a0<span id=\"MathJax-Element-7-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-188\" class=\"mjx-math\"><span id=\"MJXc-Node-189\" class=\"mjx-mrow\"><span id=\"MJXc-Node-190\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03bc<\/span><\/span><span id=\"MJXc-Node-191\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-192\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-193\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-195\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-196\" class=\"mjx-mi MJXc-space1\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c3<\/span><\/span><span id=\"MJXc-Node-197\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-198\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-199\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><\/span><\/span><\/p>\n<ol class=\"lower-alpha\">\n<li>\n<p id=\"N10DD8\">Find Q1, the median, and Q3. Q1 = 266 \u2013 .67(16) = 255; median = mean = 266;<\/p>\n<p id=\"N10DDB\">Q3 = 266 + .67(16) = 277. Thus, the probability is 1\/4 that a pregnancy will last less than 255 days; 1\/2 that it will last less than 266 days; 3\/4 that it will last less than 277 days.<\/p>\n<p><span class=\"imagewrap\"><span class=\"image\"><img decoding=\"async\" id=\"_i_5\" class=\"img-responsive popimg aligncenter\" title=\"A normal probability distribution curve on which the horizontal axis is labeled &quot;preg.days x.&quot; The first quartile is marked at 255, with .25 area to the left of it under the curve. The median is marked at 266, with .50 area to the left of it. Lastly, the third quartile is marked at 277, with .75 area to the left of it. In between each quartile is .25 of area.\" src=\"https:\/\/oli.cmu.edu\/repository\/webcontent\/72712ec00a0001dc418a87e73e8ebb77\/_u4_probability\/_m3_random_variables\/webcontent\/image162.gif\" alt=\"A normal probability distribution curve on which the horizontal axis is labeled &quot;preg.days x.&quot; The first quartile is marked at 255, with .25 area to the left of it under the curve. The median is marked at 266, with .50 area to the left of it. Lastly, the third quartile is marked at 277, with .75 area to the left of it. In between each quartile is .25 of area.\" \/><\/span><\/span><\/li>\n<li>\n<p id=\"N10DE5\">What is the probability that a randomly chosen pregnancy will last less than 246 days? Since (246 \u2013 266) \/ 16 = -1.25, we write<\/p>\n<p id=\"N10DE8\"><span id=\"MathJax-Element-8-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-200\" class=\"mjx-math\"><span id=\"MJXc-Node-201\" class=\"mjx-mrow\"><span id=\"MJXc-Node-202\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-203\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-204\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-205\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-206\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-207\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-208\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-209\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-210\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-211\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-212\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-213\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-214\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-215\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-216\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-217\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-218\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-219\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-220\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-221\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-222\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-223\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-224\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-225\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-226\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-227\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li>\n<p id=\"N10E3E\">What is the probability that a randomly chosen pregnancy will last longer than 240 days? Since (240 \u2013 266) \/ 16 = -1.63, we write<\/p>\n<p id=\"N10E41\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-228\" class=\"mjx-math\"><span id=\"MJXc-Node-229\" class=\"mjx-mrow\"><span id=\"MJXc-Node-230\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-231\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-232\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-233\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-234\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-235\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-236\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-237\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-238\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-239\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-240\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-241\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-242\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-243\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-244\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-245\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-246\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-247\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-248\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-249\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-250\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-251\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-252\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-253\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-254\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-255\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-256\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-257\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-258\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-259\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-260\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-261\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-262\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-263\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-264\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-265\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-266\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/p>\n<p id=\"N10EB7\">Since the mean is 266 and the standard deviation is 16, most pregnancies last longer than 240 days.<\/p>\n<\/li>\n<li>\n<p id=\"N10EBB\">What is the probability that a randomly chosen pregnancy will last longer than 500 days?<\/p>\n<p id=\"N10EBE\"><em>Method 1:<\/em>\u00a0Common sense tells us that this would be impossible.<\/p>\n<p id=\"N10EC3\"><em>Method 2:<\/em>\u00a0The standardized value of 500 is (500 \u2013 266) \/ 16 = +14.625.\u00a0<span id=\"MathJax-Element-10-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-267\" class=\"mjx-math\"><span id=\"MJXc-Node-268\" class=\"mjx-mrow\"><span id=\"MJXc-Node-269\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-270\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-271\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-272\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-273\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-274\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-275\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-276\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-277\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-278\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-279\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-280\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-281\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-282\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-283\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-284\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-285\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-286\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-287\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-288\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-289\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-290\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<li>\n<p id=\"N10F0F\">Suppose a pregnant woman\u2019s husband has scheduled his business trips so that he will be in town between the 235th and 295th days. What is the probability that the birth will take place during that time? The standardized values are (235 \u2013 266) \/ 16) = -1.94 and (295 \u2013 266) \/ 16 = +1.81.<\/p>\n<p id=\"N10F12\"><span id=\"MathJax-Element-11-Frame\" class=\"mjx-chtml MathJax_CHTML\"><span id=\"MJXc-Node-291\" class=\"mjx-math\"><span id=\"MJXc-Node-292\" class=\"mjx-mrow\"><span id=\"MJXc-Node-293\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-294\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-295\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-296\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-297\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-298\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-299\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">X<\/span><\/span><span id=\"MJXc-Node-300\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-301\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-302\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-303\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-304\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-305\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-306\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-307\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-308\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-309\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-310\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-311\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-312\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-313\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-314\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-315\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-316\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-317\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-318\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-319\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-320\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-321\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-322\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-323\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-324\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-325\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-326\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-327\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-328\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-329\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-330\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-331\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-332\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-333\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-334\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-335\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-336\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">Z<\/span><\/span><span id=\"MJXc-Node-337\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-338\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-339\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-340\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-341\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-342\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-343\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><span id=\"MJXc-Node-344\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-345\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-346\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-347\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-348\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-349\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><span id=\"MJXc-Node-350\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-351\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-352\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-353\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-354\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-355\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-356\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-357\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-358\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-359\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><span id=\"MJXc-Node-360\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">.<\/span><\/span><span id=\"MJXc-Node-361\" class=\"mjx-mn MJXc-space1\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><span id=\"MJXc-Node-362\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-363\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><span id=\"MJXc-Node-364\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7<\/span><\/span><\/span><\/span><\/span><\/p>\n<p id=\"N10FF1\">There is close to a 94% chance that the husband will be in town for the birth.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<h3 class=\"textbox__title\">Learn by Doing<\/h3>\n<\/header>\n<div class=\"textbox__content\">\n<p>According to the College Board website, the scores on the math part of the SAT (SAT-M) in a certain year had a mean of 507 and standard deviation of 111. Assume that SAT scores follow a normal distribution.<\/p>\n<div id=\"h5p-136\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-136\" class=\"h5p-iframe\" data-content-id=\"136\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"6.3 learn by doing 1\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":150,"menu_order":14,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-521","chapter","type-chapter","status-publish","hentry","chapter-type-numberless"],"part":419,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapters\/521","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/wp\/v2\/users\/150"}],"version-history":[{"count":10,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapters\/521\/revisions"}],"predecessor-version":[{"id":984,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapters\/521\/revisions\/984"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/parts\/419"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapters\/521\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/wp\/v2\/media?parent=521"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/pressbooks\/v2\/chapter-type?post=521"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/wp\/v2\/contributor?post=521"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/mat1260\/wp-json\/wp\/v2\/license?post=521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}