{"id":305,"date":"2022-05-18T16:38:24","date_gmt":"2022-05-18T16:38:24","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/central-limit-theorem-cookie-recipes\/"},"modified":"2022-11-09T16:19:41","modified_gmt":"2022-11-09T16:19:41","slug":"central-limit-theorem-cookie-recipes","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/central-limit-theorem-cookie-recipes\/","title":{"raw":"Activity 8.4: Central Limit Theorem (Cookie Recipes)","rendered":"Activity 8.4: Central Limit Theorem (Cookie Recipes)"},"content":{"raw":"&nbsp;\r\n<div id=\"fs-id1171897709330\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div data-type=\"title\">Central Limit Theorem (Cookie Recipes)<\/div>\r\n<p id=\"id5609695\">Class Time:<\/p>\r\n<p id=\"id5609702\">Names:<\/p>\r\n\r\n<div id=\"list-2397864897\" data-type=\"list\">\r\n<div data-type=\"title\">Student Learning Outcomes<\/div>\r\n<ul>\r\n \t<li>The student will demonstrate and compare properties of the central limit theorem.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<span data-type=\"title\">Given<\/span><em data-effect=\"italics\">X<\/em> = length of time (in days) that a cookie recipe lasted at the Olmstead Homestead. (Assume that each of the different recipes makes the same quantity of cookies.)\r\n<table id=\"tableones23\" summary=\"This table presents recipe data in the first column and X length of days data in the second column of 15 rows. These two columns repeat three more times in the same table.\">\r\n<thead>\r\n<tr>\r\n<th data-align=\"center\">Recipe #<\/th>\r\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\r\n<th><\/th>\r\n<th data-align=\"center\">Recipe #<\/th>\r\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\r\n<th><\/th>\r\n<th data-align=\"center\">Recipe #<\/th>\r\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\r\n<th><\/th>\r\n<th data-align=\"center\">Recipe #<\/th>\r\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">16<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">46<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">17<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">32<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">47<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">18<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">33<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">48<\/td>\r\n<td data-align=\"center\">11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">19<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">34<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">49<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">20<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">35<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">50<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">21<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">36<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">51<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">22<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">37<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">52<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">23<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">38<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">53<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">9<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">24<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">39<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">54<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">25<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">40<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">55<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">11<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">26<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">41<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">56<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">12<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">27<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">42<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">57<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">13<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">28<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">43<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">58<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">14<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">29<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">44<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">59<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">15<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">45<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td><\/td>\r\n<td data-align=\"center\">60<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"element-538\">Calculate the following:<\/p>\r\n\r\n<ol id=\"list-2396953\" type=\"a\">\r\n \t<li><em data-effect=\"italics\">\u03bc<sub>x<\/sub><\/em> = _______<\/li>\r\n \t<li><em data-effect=\"italics\">\u03c3<sub>x<\/sub><\/em> = _______<\/li>\r\n<\/ol>\r\n<p id=\"element-453\" class=\"finger\"><span data-type=\"title\">Collect the Data<\/span>Use a random number generator to randomly select four samples of size <em data-effect=\"italics\">n<\/em> = 5 from the given population. Record your samples in <a class=\"autogenerated-content\" href=\"#Ch07_lab2_tbl002\">(Figure)<\/a>. Then, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.<\/p>\r\n\r\n<ol id=\"list-98698565\">\r\n \t<li>Complete the table:\r\n<table id=\"Ch07_lab2_tbl002\" summary=\"Partially filled table with samples (4) in columns and 5 blank rows plus a blank Means row.\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Sample 1<\/th>\r\n<th>Sample 2<\/th>\r\n<th>Sample 3<\/th>\r\n<th>Sample 4<\/th>\r\n<th>Sample means from other groups:<\/th>\r\n<\/tr>\r\n<\/thead>\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>Means:<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Calculate the following:\r\n<ol id=\"eip-idp136101696\" type=\"a\">\r\n \t<li>\\(\\overline{x}\\) = _______<\/li>\r\n \t<li><em data-effect=\"italics\">s<\/em><sub>\\(\\overline{x}\\)<\/sub> = _______<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li class=\"finger\">Again, use a random number generator to randomly select four samples from the population. This time, make the samples of size <em data-effect=\"italics\">n<\/em> = 10. Record the samples in <a class=\"autogenerated-content\" href=\"#Ch07_lab2_tbl003\">(Figure)<\/a>. As before, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.\r\n<table id=\"Ch07_lab2_tbl003\" summary=\"Same as the above table except with 10 blank rows and a means row.\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Sample 1<\/th>\r\n<th>Sample 2<\/th>\r\n<th>Sample 3<\/th>\r\n<th>Sample 4<\/th>\r\n<th>Sample means from other groups<\/th>\r\n<\/tr>\r\n<\/thead>\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<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>Means:<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td>\\(\\overline{x}\\) = ____<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Calculate the following:\r\n<ol id=\"element-8475643\" type=\"a\">\r\n \t<li>\\(\\overline{x}\\) = ______<\/li>\r\n \t<li><em data-effect=\"italics\">s<\/em><sub>\\(\\overline{x}\\)<\/sub> = ______<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>For the original population, construct a histogram. Make intervals with a bar width of one day. Sketch the graph using a ruler and pencil. Scale the axes.<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"id6597795\" class=\"bc-figure figure\"><span id=\"id6597799\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/05\/fig-ch07_10_01-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div><\/li>\r\n \t<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\r\n<\/ol>\r\n<div id=\"list-2646\" data-type=\"list\">\r\n<div data-type=\"title\">Repeat the Procedure for <em data-effect=\"italics\">n<\/em> = 5<\/div>\r\n<ol>\r\n \t<li>For the sample of <em data-effect=\"italics\">n<\/em> = 5 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \\(\\frac{1}{2}\\) a day. Sketch the graph using a ruler and pencil. Scale the axes.\r\n<div id=\"id6597892\" class=\"bc-figure figure\"><span id=\"id6597897\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/08\/fig-ch07_10_02-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div><\/li>\r\n \t<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"list-2764646\" data-type=\"list\">\r\n<div data-type=\"title\">Repeat the Procedure for <em data-effect=\"italics\">n<\/em> = 10<\/div>\r\n<ol>\r\n \t<li>For the sample of <em data-effect=\"italics\">n<\/em> = 10 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \\(\\frac{1}{2}\\) a day. Sketch the graph using a ruler and pencil. Scale the axes.\r\n<div id=\"id6597991\" class=\"bc-figure figure\"><span id=\"id6597997\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/08\/fig-ch07_10_03-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div><\/li>\r\n \t<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"list-2937597532\" data-type=\"list\">\r\n<div data-type=\"title\">Discussion Questions<\/div>\r\n<ol data-mark-suffix=\".\">\r\n \t<li>Compare the three histograms you have made, the one for the population and the two for the sample means. In three to five sentences, describe the similarities and differences.<\/li>\r\n \t<li>State the theoretical (according to the clt) distributions for the sample means.\r\n<ol id=\"list-239875697325\" type=\"a\">\r\n \t<li><em data-effect=\"italics\">n<\/em> = 5: \\(\\overline{x}\\) ~ _____(_____,_____)<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> = 10: \\(\\overline{x}\\) ~ _____(_____,_____)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Are the sample means for <em data-effect=\"italics\">n<\/em> = 5 and <em data-effect=\"italics\">n<\/em> = 10 \u201cclose\u201d to the theoretical mean, <em data-effect=\"italics\">\u03bc<sub>x<\/sub><\/em>? Explain why or why not.<\/li>\r\n \t<li>Which of the two distributions of sample means has the smaller standard deviation? Why?<\/li>\r\n \t<li>As <em data-effect=\"italics\">n<\/em> changed, why did the shape of the distribution of the data change? Use one to two complete sentences to explain what happened.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"<p>&nbsp;<\/p>\n<div id=\"fs-id1171897709330\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\">Central Limit Theorem (Cookie Recipes)<\/div>\n<p id=\"id5609695\">Class Time:<\/p>\n<p id=\"id5609702\">Names:<\/p>\n<div id=\"list-2397864897\" data-type=\"list\">\n<div data-type=\"title\">Student Learning Outcomes<\/div>\n<ul>\n<li>The student will demonstrate and compare properties of the central limit theorem.<\/li>\n<\/ul>\n<\/div>\n<p><span data-type=\"title\">Given<\/span><em data-effect=\"italics\">X<\/em> = length of time (in days) that a cookie recipe lasted at the Olmstead Homestead. (Assume that each of the different recipes makes the same quantity of cookies.)<\/p>\n<table id=\"tableones23\" summary=\"This table presents recipe data in the first column and X length of days data in the second column of 15 rows. These two columns repeat three more times in the same table.\">\n<thead>\n<tr>\n<th data-align=\"center\">Recipe #<\/th>\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\n<th><\/th>\n<th data-align=\"center\">Recipe #<\/th>\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\n<th><\/th>\n<th data-align=\"center\">Recipe #<\/th>\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\n<th><\/th>\n<th data-align=\"center\">Recipe #<\/th>\n<th data-align=\"center\"><em data-effect=\"italics\">X<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td data-align=\"center\">46<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">17<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">32<\/td>\n<td data-align=\"center\">4<\/td>\n<td><\/td>\n<td data-align=\"center\">47<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">18<\/td>\n<td data-align=\"center\">4<\/td>\n<td><\/td>\n<td data-align=\"center\">33<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">48<\/td>\n<td data-align=\"center\">11<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">19<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">34<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">49<\/td>\n<td data-align=\"center\">5<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">35<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">50<\/td>\n<td data-align=\"center\">5<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">21<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">36<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">51<\/td>\n<td data-align=\"center\">4<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">22<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">37<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">52<\/td>\n<td data-align=\"center\">6<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">23<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">38<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">53<\/td>\n<td data-align=\"center\">5<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">9<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">24<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">39<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">54<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">25<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">40<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">55<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">11<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td data-align=\"center\">26<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">41<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">56<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">12<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">27<\/td>\n<td data-align=\"center\">4<\/td>\n<td><\/td>\n<td data-align=\"center\">42<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">57<\/td>\n<td data-align=\"center\">4<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">13<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">28<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td data-align=\"center\">43<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">58<\/td>\n<td data-align=\"center\">3<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">14<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td data-align=\"center\">29<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">44<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td data-align=\"center\">59<\/td>\n<td data-align=\"center\">6<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">15<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">45<\/td>\n<td data-align=\"center\">2<\/td>\n<td><\/td>\n<td data-align=\"center\">60<\/td>\n<td data-align=\"center\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"element-538\">Calculate the following:<\/p>\n<ol id=\"list-2396953\" type=\"a\">\n<li><em data-effect=\"italics\">\u03bc<sub>x<\/sub><\/em> = _______<\/li>\n<li><em data-effect=\"italics\">\u03c3<sub>x<\/sub><\/em> = _______<\/li>\n<\/ol>\n<p id=\"element-453\" class=\"finger\"><span data-type=\"title\">Collect the Data<\/span>Use a random number generator to randomly select four samples of size <em data-effect=\"italics\">n<\/em> = 5 from the given population. Record your samples in <a class=\"autogenerated-content\" href=\"#Ch07_lab2_tbl002\">(Figure)<\/a>. Then, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.<\/p>\n<ol id=\"list-98698565\">\n<li>Complete the table:<br \/>\n<table id=\"Ch07_lab2_tbl002\" summary=\"Partially filled table with samples (4) in columns and 5 blank rows plus a blank Means row.\">\n<thead>\n<tr>\n<th><\/th>\n<th>Sample 1<\/th>\n<th>Sample 2<\/th>\n<th>Sample 3<\/th>\n<th>Sample 4<\/th>\n<th>Sample means from other groups:<\/th>\n<\/tr>\n<\/thead>\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>Means:<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calculate the following:\n<ol id=\"eip-idp136101696\" type=\"a\">\n<li>\\(\\overline{x}\\) = _______<\/li>\n<li><em data-effect=\"italics\">s<\/em><sub>\\(\\overline{x}\\)<\/sub> = _______<\/li>\n<\/ol>\n<\/li>\n<li class=\"finger\">Again, use a random number generator to randomly select four samples from the population. This time, make the samples of size <em data-effect=\"italics\">n<\/em> = 10. Record the samples in <a class=\"autogenerated-content\" href=\"#Ch07_lab2_tbl003\">(Figure)<\/a>. As before, for each sample, calculate the mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest of the class.<br \/>\n<table id=\"Ch07_lab2_tbl003\" summary=\"Same as the above table except with 10 blank rows and a means row.\">\n<thead>\n<tr>\n<th><\/th>\n<th>Sample 1<\/th>\n<th>Sample 2<\/th>\n<th>Sample 3<\/th>\n<th>Sample 4<\/th>\n<th>Sample means from other groups<\/th>\n<\/tr>\n<\/thead>\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<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>Means:<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td>\\(\\overline{x}\\) = ____<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calculate the following:\n<ol id=\"element-8475643\" type=\"a\">\n<li>\\(\\overline{x}\\) = ______<\/li>\n<li><em data-effect=\"italics\">s<\/em><sub>\\(\\overline{x}\\)<\/sub> = ______<\/li>\n<\/ol>\n<\/li>\n<li>For the original population, construct a histogram. Make intervals with a bar width of one day. Sketch the graph using a ruler and pencil. Scale the axes.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"id6597795\" class=\"bc-figure figure\"><span id=\"id6597799\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/05\/fig-ch07_10_01-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div>\n<\/li>\n<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\n<\/ol>\n<div id=\"list-2646\" data-type=\"list\">\n<div data-type=\"title\">Repeat the Procedure for <em data-effect=\"italics\">n<\/em> = 5<\/div>\n<ol>\n<li>For the sample of <em data-effect=\"italics\">n<\/em> = 5 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \\(\\frac{1}{2}\\) a day. Sketch the graph using a ruler and pencil. Scale the axes.\n<div id=\"id6597892\" class=\"bc-figure figure\"><span id=\"id6597897\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/08\/fig-ch07_10_02-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div>\n<\/li>\n<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\n<\/ol>\n<\/div>\n<div id=\"list-2764646\" data-type=\"list\">\n<div data-type=\"title\">Repeat the Procedure for <em data-effect=\"italics\">n<\/em> = 10<\/div>\n<ol>\n<li>For the sample of <em data-effect=\"italics\">n<\/em> = 10 days averaged together, construct a histogram of the averages (your means together with the means of the other groups). Make intervals with bar widths of \\(\\frac{1}{2}\\) a day. Sketch the graph using a ruler and pencil. Scale the axes.\n<div id=\"id6597991\" class=\"bc-figure figure\"><span id=\"id6597997\" data-type=\"media\" data-alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.ccconline.org\/acccomposition1\/wp-content\/uploads\/sites\/83\/2022\/08\/fig-ch07_10_03-1.png\" alt=\"This is a blank graph template. The horizontal axis is labeled Time (days) and the vertical axis is labeled Frequency.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/div>\n<\/li>\n<li>Draw a smooth curve through the tops of the bars of the histogram. Use one to two complete sentences to describe the general shape of the curve.<\/li>\n<\/ol>\n<\/div>\n<div id=\"list-2937597532\" data-type=\"list\">\n<div data-type=\"title\">Discussion Questions<\/div>\n<ol data-mark-suffix=\".\">\n<li>Compare the three histograms you have made, the one for the population and the two for the sample means. In three to five sentences, describe the similarities and differences.<\/li>\n<li>State the theoretical (according to the clt) distributions for the sample means.\n<ol id=\"list-239875697325\" type=\"a\">\n<li><em data-effect=\"italics\">n<\/em> = 5: \\(\\overline{x}\\) ~ _____(_____,_____)<\/li>\n<li><em data-effect=\"italics\">n<\/em> = 10: \\(\\overline{x}\\) ~ _____(_____,_____)<\/li>\n<\/ol>\n<\/li>\n<li>Are the sample means for <em data-effect=\"italics\">n<\/em> = 5 and <em data-effect=\"italics\">n<\/em> = 10 \u201cclose\u201d to the theoretical mean, <em data-effect=\"italics\">\u03bc<sub>x<\/sub><\/em>? Explain why or why not.<\/li>\n<li>Which of the two distributions of sample means has the smaller standard deviation? Why?<\/li>\n<li>As <em data-effect=\"italics\">n<\/em> changed, why did the shape of the distribution of the data change? Use one to two complete sentences to explain what happened.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n","protected":false},"author":32,"menu_order":48,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-305","chapter","type-chapter","status-publish","hentry"],"part":277,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/305","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\/305\/revisions"}],"predecessor-version":[{"id":675,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/305\/revisions\/675"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/parts\/277"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/305\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/media?parent=305"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapter-type?post=305"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/contributor?post=305"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/license?post=305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}