{"id":1154,"date":"2023-06-22T01:56:54","date_gmt":"2023-06-22T01:56:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-learn-it-3\/"},"modified":"2024-10-21T15:55:30","modified_gmt":"2024-10-21T15:55:30","slug":"sampling-variability-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-learn-it-3\/","title":{"raw":"Sampling Variability: Learn It 3","rendered":"Sampling Variability: Learn It 3"},"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<h2>Calculating Probabilities for the Sampling Distribution of the Sample Proportion<\/h2>\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 <strong>mean<\/strong> of the distribution of sample proportions is [latex]p[\/latex].<\/li>\r\n\t<li>The <strong>standard deviation<\/strong> of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\r\n\t<li>The <strong>normal condition<\/strong> states that if [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex] then the sampling distribution is approximately normal by the Central Limit Theorem.\u00a0<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" 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]1604[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">When the sample size is large enough, we can use [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}} [\/latex] in place of[latex] \\sqrt{\\frac{p(1-p)}{n}} [\/latex].\u00a0 This is called the <strong>standard error<\/strong>, which is the estimated standard deviation of sample proportions.<\/section>\r\n<\/section>","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<h2>Calculating Probabilities for the Sampling Distribution of the Sample Proportion<\/h2>\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 <strong>mean<\/strong> of the distribution of sample proportions is [latex]p[\/latex].<\/li>\n<li>The <strong>standard deviation<\/strong> of the distribution of sample proportions is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/li>\n<li>The <strong>normal condition<\/strong> states that if [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex] then the sampling distribution is approximately normal by the Central Limit Theorem.\u00a0<\/li>\n<\/ul>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1604\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1604&theme=lumen&iframe_resize_id=ohm1604&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\">When the sample size is large enough, we can use [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex] in place of[latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].\u00a0 This is called the <strong>standard error<\/strong>, which is the estimated standard deviation of sample proportions.<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":22,"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":"learn_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\/1154"}],"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":16,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1154\/revisions"}],"predecessor-version":[{"id":6136,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1154\/revisions\/6136"}],"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\/1154\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1154"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1154"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1154"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}