{"id":272,"date":"2022-05-18T16:38:11","date_gmt":"2022-05-18T16:38:11","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/normal-distribution-lap-times\/"},"modified":"2022-11-09T16:15:01","modified_gmt":"2022-11-09T16:15:01","slug":"normal-distribution-lap-times","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/normal-distribution-lap-times\/","title":{"raw":"Activity 7.4: Normal Distribution (Lap Times)","rendered":"Activity 7.4: Normal Distribution (Lap Times)"},"content":{"raw":"&nbsp;\r\n<div id=\"fs-id8352229\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div data-type=\"title\">Normal Distribution (Lap Times)<\/div>\r\n<p id=\"id5906467\">Class Time:<\/p>\r\n<p id=\"id7315926\">Names:<\/p>\r\n\r\n<div id=\"id3819098\" data-type=\"list\">\r\n<div data-type=\"title\">Student Learning Outcome<\/div>\r\n<ul>\r\n \t<li>The student will compare and contrast empirical data and a theoretical distribution to determine if Terry Vogel's lap times fit a continuous distribution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<span data-type=\"title\">Directions<\/span>Round the relative frequencies and probabilities to four decimal places. Carry all other decimal answers to two places.\r\n<div id=\"list-23435\" data-type=\"list\">\r\n<div id=\"CollectData\" data-type=\"title\">Collect the Data<\/div>\r\n<ol>\r\n \t<li>Use the data from <a href=\"\/contents\/3ef830bc-5247-460a-9007-e3fd762e5e93\">Appendix C<\/a>. Use a stratified sampling method by lap (races 1 to 20) and a random number generator to pick six lap times from each stratum. Record the lap times below for laps two to seven.<span data-type=\"newline\">\r\n<\/span>\r\n<table id=\"element-2352564535\" summary=\"Blank table with 36 empty cells.\">\r\n<tbody>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<td>_______<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil. Scale the axes.\r\n<div id=\"fig045324\" class=\"bc-figure figure\"><span id=\"id7826845\" data-type=\"media\" data-alt=\"Blank graph with relative frequency on the vertical axis and lap time on the horizontal axis.\"><img src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/05\/fig-ch06_10_01-1.png\" alt=\"Blank graph with relative frequency on the vertical axis and lap time on the horizontal axis.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/div><\/li>\r\n \t<li>Calculate the following:\r\n<ol id=\"list-245243533\" type=\"a\">\r\n \t<li>\\(\\overline{x}\\) = _______<\/li>\r\n \t<li><em data-effect=\"italics\">s<\/em> = _______<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw a smooth curve through the tops of the bars of the histogram. Write one to two complete sentences to describe the general shape of the curve. (Keep it simple. Does the graph go straight across, does it have a v-shape, does it have a hump in the middle or at either end, and so on?)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<span id=\"AnalyzeDist\" data-type=\"title\">Analyze the Distribution<\/span> Using your sample mean, sample standard deviation, and histogram to help, what is the approximate theoretical distribution of the data?\r\n<ul id=\"list-23452\">\r\n \t<li><em data-effect=\"italics\">X<\/em> ~ _____(_____,_____)<\/li>\r\n \t<li>How does the histogram help you arrive at the approximate distribution?<\/li>\r\n<\/ul>\r\n<span id=\"DescData\" data-type=\"title\">Describe the Data<\/span> Use the data you collected to complete the following statements.\r\n<ul id=\"list-2342532\">\r\n \t<li>The <em data-effect=\"italics\">IQR<\/em> goes from __________ to __________.<\/li>\r\n \t<li><em data-effect=\"italics\">IQR<\/em> = __________. (<em data-effect=\"italics\">IQR<\/em> = <em data-effect=\"italics\">Q<\/em><sub>3<\/sub> \u2013 <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>)<\/li>\r\n \t<li>The 15<sup>th<\/sup> percentile is _______.<\/li>\r\n \t<li>The 85<sup>th<\/sup> percentile is _______.<\/li>\r\n \t<li>The median is _______.<\/li>\r\n \t<li>The empirical probability that a randomly chosen lap time is more than 130 seconds is _______.<\/li>\r\n \t<li>Explain the meaning of the 85<sup>th<\/sup> percentile of this data.<\/li>\r\n<\/ul>\r\n<span data-type=\"title\">Theoretical Distribution<\/span> Using the theoretical distribution, complete the following statements. You should use a normal approximation based on your sample data.\r\n<ul id=\"list-23432432\">\r\n \t<li>The <em data-effect=\"italics\">IQR<\/em> goes from __________ to __________.<\/li>\r\n \t<li><em data-effect=\"italics\">IQR<\/em> = _______.<\/li>\r\n \t<li>The 15<sup>th<\/sup> percentile is _______.<\/li>\r\n \t<li>The 85<sup>th<\/sup> percentile is _______.<\/li>\r\n \t<li>The median is _______.<\/li>\r\n \t<li>The probability that a randomly chosen lap time is more than 130 seconds is _______.<\/li>\r\n \t<li>Explain the meaning of the 85<sup>th<\/sup> percentile of this distribution.<\/li>\r\n<\/ul>\r\n<p id=\"fs-idm197255200\"><span data-type=\"title\">Discussion Questions<\/span>Do the data from the section titled <a href=\"#CollectData\">Collect the Data<\/a> give a close approximation to the theoretical distribution in the section titled <a href=\"#AnalyzeDist\">Analyze the Distribution<\/a>? In complete sentences and comparing the result in the sections titled <a href=\"#DescData\">Describe the Data<\/a> and <a href=\"#TheoDist\">Theoretical Distribution<\/a>, explain why or why not.<\/p>\r\n\r\n<\/div>","rendered":"<p>&nbsp;<\/p>\n<div id=\"fs-id8352229\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\">Normal Distribution (Lap Times)<\/div>\n<p id=\"id5906467\">Class Time:<\/p>\n<p id=\"id7315926\">Names:<\/p>\n<div id=\"id3819098\" data-type=\"list\">\n<div data-type=\"title\">Student Learning Outcome<\/div>\n<ul>\n<li>The student will compare and contrast empirical data and a theoretical distribution to determine if Terry Vogel&#8217;s lap times fit a continuous distribution.<\/li>\n<\/ul>\n<\/div>\n<p><span data-type=\"title\">Directions<\/span>Round the relative frequencies and probabilities to four decimal places. Carry all other decimal answers to two places.<\/p>\n<div id=\"list-23435\" data-type=\"list\">\n<div id=\"CollectData\" data-type=\"title\">Collect the Data<\/div>\n<ol>\n<li>Use the data from <a href=\"\/contents\/3ef830bc-5247-460a-9007-e3fd762e5e93\">Appendix C<\/a>. Use a stratified sampling method by lap (races 1 to 20) and a random number generator to pick six lap times from each stratum. Record the lap times below for laps two to seven.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"element-2352564535\" summary=\"Blank table with 36 empty cells.\">\n<tbody>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<tr>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<td>_______<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil. Scale the axes.\n<div id=\"fig045324\" class=\"bc-figure figure\"><span id=\"id7826845\" data-type=\"media\" data-alt=\"Blank graph with relative frequency on the vertical axis and lap time on the horizontal axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/05\/fig-ch06_10_01-1.png\" alt=\"Blank graph with relative frequency on the vertical axis and lap time on the horizontal axis.\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/div>\n<\/li>\n<li>Calculate the following:\n<ol id=\"list-245243533\" type=\"a\">\n<li>\\(\\overline{x}\\) = _______<\/li>\n<li><em data-effect=\"italics\">s<\/em> = _______<\/li>\n<\/ol>\n<\/li>\n<li>Draw a smooth curve through the tops of the bars of the histogram. Write one to two complete sentences to describe the general shape of the curve. (Keep it simple. Does the graph go straight across, does it have a v-shape, does it have a hump in the middle or at either end, and so on?)<\/li>\n<\/ol>\n<\/div>\n<p><span id=\"AnalyzeDist\" data-type=\"title\">Analyze the Distribution<\/span> Using your sample mean, sample standard deviation, and histogram to help, what is the approximate theoretical distribution of the data?<\/p>\n<ul id=\"list-23452\">\n<li><em data-effect=\"italics\">X<\/em> ~ _____(_____,_____)<\/li>\n<li>How does the histogram help you arrive at the approximate distribution?<\/li>\n<\/ul>\n<p><span id=\"DescData\" data-type=\"title\">Describe the Data<\/span> Use the data you collected to complete the following statements.<\/p>\n<ul id=\"list-2342532\">\n<li>The <em data-effect=\"italics\">IQR<\/em> goes from __________ to __________.<\/li>\n<li><em data-effect=\"italics\">IQR<\/em> = __________. (<em data-effect=\"italics\">IQR<\/em> = <em data-effect=\"italics\">Q<\/em><sub>3<\/sub> \u2013 <em data-effect=\"italics\">Q<\/em><sub>1<\/sub>)<\/li>\n<li>The 15<sup>th<\/sup> percentile is _______.<\/li>\n<li>The 85<sup>th<\/sup> percentile is _______.<\/li>\n<li>The median is _______.<\/li>\n<li>The empirical probability that a randomly chosen lap time is more than 130 seconds is _______.<\/li>\n<li>Explain the meaning of the 85<sup>th<\/sup> percentile of this data.<\/li>\n<\/ul>\n<p><span data-type=\"title\">Theoretical Distribution<\/span> Using the theoretical distribution, complete the following statements. You should use a normal approximation based on your sample data.<\/p>\n<ul id=\"list-23432432\">\n<li>The <em data-effect=\"italics\">IQR<\/em> goes from __________ to __________.<\/li>\n<li><em data-effect=\"italics\">IQR<\/em> = _______.<\/li>\n<li>The 15<sup>th<\/sup> percentile is _______.<\/li>\n<li>The 85<sup>th<\/sup> percentile is _______.<\/li>\n<li>The median is _______.<\/li>\n<li>The probability that a randomly chosen lap time is more than 130 seconds is _______.<\/li>\n<li>Explain the meaning of the 85<sup>th<\/sup> percentile of this distribution.<\/li>\n<\/ul>\n<p id=\"fs-idm197255200\"><span data-type=\"title\">Discussion Questions<\/span>Do the data from the section titled <a href=\"#CollectData\">Collect the Data<\/a> give a close approximation to the theoretical distribution in the section titled <a href=\"#AnalyzeDist\">Analyze the Distribution<\/a>? In complete sentences and comparing the result in the sections titled <a href=\"#DescData\">Describe the Data<\/a> and <a href=\"#TheoDist\">Theoretical Distribution<\/a>, explain why or why not.<\/p>\n<\/div>\n","protected":false},"author":32,"menu_order":43,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-272","chapter","type-chapter","status-publish","hentry"],"part":250,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/272","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/users\/32"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/272\/revisions"}],"predecessor-version":[{"id":667,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/272\/revisions\/667"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/parts\/250"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/272\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/media?parent=272"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapter-type?post=272"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/contributor?post=272"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/license?post=272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}