{"id":1157,"date":"2023-06-22T01:56:57","date_gmt":"2023-06-22T01:56:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-apply-it-2\/"},"modified":"2024-02-29T20:01:05","modified_gmt":"2024-02-29T20:01:05","slug":"sampling-variability-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-apply-it-2\/","title":{"raw":"Sampling Variability: Apply It 2","rendered":"Sampling Variability: Apply It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions for normal approximation of a sampling distribution of a sample proportion&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions for normal approximation of a sampling distribution of a sample proportion.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal distribution to calculate probabilities and percentiles from a sampling distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal distribution to calculate probabilities and percentiles from a sampling distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find the sample size needed for a sampling distribution to have a desired standard deviation.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Let's use the statistical tool below to simulate samples of different sizes from the American adult population, where the sample proportion who are obese is calculated for each sample.<\/p>\r\n<section class=\"textbox interact\">Let's conduct a study to decide if there is evidence that the rate of obesity among the clinic's patients is less than [latex]42.4\\%[\/latex].You will need to check the <strong>Enter Numerical Values for [latex]n[\/latex] and [latex]p[\/latex]<\/strong>\u00a0box to enter the value [latex]n[\/latex] and [latex]p[\/latex].Observe the center, spread, and shape of the sampling distribution as the sample size increases as instructed below.\r\n\r\n<ul>\r\n\t<li>Set the sample size to [latex]n=1[\/latex]. Then draw [latex]1,000[\/latex] random samples of size [latex]1[\/latex] from the population.<\/li>\r\n\t<li>Set the sample size to [latex]n=5[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]5[\/latex] from the population.<\/li>\r\n\t<li>Set the sample size to [latex]n=25[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]25[\/latex] from the population.<\/li>\r\n\t<li>Set the sample size to [latex]n=100[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]100[\/latex] from the population.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_prop\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_prop\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1595[\/ohm2_question]<\/section>\r\n<div>\r\n<p>In the question above, you\u00a0witnessed the <strong>Central Limit Theorem<\/strong> at work.<\/p>\r\n<section class=\"textbox recall\">The <strong>Central Limit Theorem<\/strong> states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>Sampling Distribution of the Sample Proportion<\/strong><\/h3>\r\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with proportion [latex]p[\/latex]:<\/p>\r\n<ul>\r\n\t<li>The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\r\n\t<li>The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\r\n\t<li>If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the <strong>Central Limit Theorem<\/strong> states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1813[\/ohm2_question]<\/section>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions for normal approximation of a sampling distribution of a sample proportion&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the conditions for normal approximation of a sampling distribution of a sample proportion.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal distribution to calculate probabilities and percentiles from a sampling distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal distribution to calculate probabilities and percentiles from a sampling distribution&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find the sample size needed for a sampling distribution to have a desired standard deviation.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>Let&#8217;s use the statistical tool below to simulate samples of different sizes from the American adult population, where the sample proportion who are obese is calculated for each sample.<\/p>\n<section class=\"textbox interact\">Let&#8217;s conduct a study to decide if there is evidence that the rate of obesity among the clinic&#8217;s patients is less than [latex]42.4\\%[\/latex].You will need to check the <strong>Enter Numerical Values for [latex]n[\/latex] and [latex]p[\/latex]<\/strong>\u00a0box to enter the value [latex]n[\/latex] and [latex]p[\/latex].Observe the center, spread, and shape of the sampling distribution as the sample size increases as instructed below.<\/p>\n<ul>\n<li>Set the sample size to [latex]n=1[\/latex]. Then draw [latex]1,000[\/latex] random samples of size [latex]1[\/latex] from the population.<\/li>\n<li>Set the sample size to [latex]n=5[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]5[\/latex] from the population.<\/li>\n<li>Set the sample size to [latex]n=25[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]25[\/latex] from the population.<\/li>\n<li>Set the sample size to [latex]n=100[\/latex] and click <strong>Reset<\/strong>. Draw [latex]1,000[\/latex] random samples of size [latex]100[\/latex] from the population.<\/li>\n<\/ul>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_prop\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_prop\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1595\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1595&theme=lumen&iframe_resize_id=ohm1595&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div>\n<p>In the question above, you\u00a0witnessed the <strong>Central Limit Theorem<\/strong> at work.<\/p>\n<section class=\"textbox recall\">The <strong>Central Limit Theorem<\/strong> states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>Sampling Distribution of the Sample Proportion<\/strong><\/h3>\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with proportion [latex]p[\/latex]:<\/p>\n<ul>\n<li>The mean of the distribution of sample proportions is [latex]p[\/latex].<\/li>\n<li>The standard deviation of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\n<li>If [latex]np\\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the <strong>Central Limit Theorem<\/strong> states that the distribution of the sample proportions follows an approximate normal distribution with mean p and standard deviation [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1813\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1813&theme=lumen&iframe_resize_id=ohm1813&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":8,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1126,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1157"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1157\/revisions"}],"predecessor-version":[{"id":5751,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1157\/revisions\/5751"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1126"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1157\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1157"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1157"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1157"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}