{"id":1155,"date":"2023-06-22T01:56:55","date_gmt":"2023-06-22T01:56:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-learn-it-4\/"},"modified":"2025-05-16T02:56:50","modified_gmt":"2025-05-16T02:56:50","slug":"sampling-variability-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-learn-it-4\/","title":{"raw":"Sampling Variability: Learn It 4","rendered":"Sampling Variability: Learn It 4"},"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>Calculate the Sample Size [latex]n[\/latex]<\/h2>\r\n<p>Recall that the standard deviation of the sampling distribution for a sample proportion can be calculated as [latex]\\sigma_{\\hat{p}}=\\sqrt{\\frac{p(1-p)}{n}}[\/latex]. If the researchers want to limit their standard deviation, or variability, then they can use this formula to find the sample size instead.<\/p>\r\n<section class=\"textbox example\">\r\n<p>[latex] \\begin{array}{l l} \\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n}} &amp;\\quad&amp; \\text{Start with the standard deviation formula. } \\\\ (\\sigma_{\\hat{p}})^2 = \\frac{p(1-p)}{n} &amp;&amp; \\text{Square both sides to eliminate the square root.} &amp;&amp;\\\\ n \\cdot (\\sigma_{\\hat{p}})^2 = p(1-p) &amp;&amp; \\text{Multiply both sides by } n \\text{ to isolate } p(1-p) \\text{ on one side.} \\\\ n = \\frac{p(1-p)}{(\\sigma_{\\hat{p}})^2} &amp;&amp; \\text{Divide both sides by } (\\sigma_{\\hat{p}})^2 \\text{ to solve for } n. \\end{array}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]1599[\/ohm2_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">[reveal-answer q=\"297770\"]See the Example Question[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"297770\"][ohm2_question hide_question_numbers=1]1383[\/ohm2_question][\/hidden-answer]<br \/>\r\n[videopicker divId=\"tnh-video-picker\" title=\"Sample Size and Normal Distribution for a Sample Proportion\" label=\"Select Instructor\"]<br \/>\r\n[videooption displayName=\"Dr. Pamela E. Harris\" value=\"https:\/\/www.youtube.com\/watch?v=tXgtPmmoO4w\"][videooption displayName=\"Dr. Aris Winger\" value=\"https:\/\/www.youtube.com\/watch?v=v8qzwEb9V1I\"] [videooption displayName=\"Dr. Lane Fisher\" value=\"https:\/\/www.youtube.com\/watch?v=6Ah-HVUEo8I\"]<br \/>\r\n[\/videopicker]<\/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>Calculate the Sample Size [latex]n[\/latex]<\/h2>\n<p>Recall that the standard deviation of the sampling distribution for a sample proportion can be calculated as [latex]\\sigma_{\\hat{p}}=\\sqrt{\\frac{p(1-p)}{n}}[\/latex]. If the researchers want to limit their standard deviation, or variability, then they can use this formula to find the sample size instead.<\/p>\n<section class=\"textbox example\">\n[latex]\\begin{array}{l l} \\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n}} &\\quad& \\text{Start with the standard deviation formula. } \\\\ (\\sigma_{\\hat{p}})^2 = \\frac{p(1-p)}{n} && \\text{Square both sides to eliminate the square root.} &&\\\\ n \\cdot (\\sigma_{\\hat{p}})^2 = p(1-p) && \\text{Multiply both sides by } n \\text{ to isolate } p(1-p) \\text{ on one side.} \\\\ n = \\frac{p(1-p)}{(\\sigma_{\\hat{p}})^2} && \\text{Divide both sides by } (\\sigma_{\\hat{p}})^2 \\text{ to solve for } n. \\end{array}[\/latex]<br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm1599\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1599&theme=lumen&iframe_resize_id=ohm1599&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox example\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q297770\">See the Example Question<\/button><\/p>\n<div id=\"q297770\" class=\"hidden-answer\" style=\"display: none\"><iframe loading=\"lazy\" id=\"ohm1383\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1383&theme=lumen&iframe_resize_id=ohm1383&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/div>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Sample Size and Normal Distribution for a Sample Proportion<\/h3>\n<form><label>Select Instructor:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/tXgtPmmoO4w\">Dr. Pamela E. Harris<\/option><option value=\"https:\/\/www.youtube.com\/embed\/v8qzwEb9V1I\">Dr. Aris Winger<\/option><option value=\"https:\/\/www.youtube.com\/embed\/6Ah-HVUEo8I\">Dr. Lane Fisher<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/tXgtPmmoO4w\" allowfullscreen><\/iframe><\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":23,"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":[{"divId":"tnh-video-picker","title":"Sample Size and Normal Distribution for a Sample Proportion","label":"Select Instructor","video_collection":[{"displayName":"Dr. Pamela E. 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