{"id":1912,"date":"2023-07-21T00:20:09","date_gmt":"2023-07-21T00:20:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=1912"},"modified":"2025-05-16T02:44:03","modified_gmt":"2025-05-16T02:44:03","slug":"sampling-distribution-of-a-sample-proportion-learn-it-4-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-proportion-learn-it-4-2\/","title":{"raw":"Sampling Distribution of a Sample Proportion: Learn It 4","rendered":"Sampling Distribution of a Sample Proportion: Learn It 4"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use technology to create a sampling distribution of a sample proportion given [latex]n[\/latex] and [latex]p[\/latex].<\/li>\r\n\t<li>Calculate the mean and standard deviation for a sampling distribution of a sample proportion.<\/li>\r\n\t<li>Recognize the difference between the standard deviation and the standard error of a sample proportion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Standard Deviation vs. Standard Error<\/h2>\r\n<p>The previous exercise assumed that we knew the value of the population proportion and could either calculate the mean and standard deviation of the sample proportion by formulas, or estimate the mean and standard deviation by simulation. In practice, we do not typically know the population proportion, nor do we have the luxury of taking thousands of random samples. Instead, we observe a single random sample. In this case, we need to <strong>estimate<\/strong> the mean and standard deviation of the sample proportion.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>The estimated mean and standard deviation of the sampling distribution of a sample proportion<\/h3>\r\n<ul>\r\n\t<li>The <strong>estimated\u00a0<\/strong>mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\r\n\t<li>To distinguish it from the true standard deviation of sample proportions, we call the <strong>estimated<\/strong> standard deviation of sample proportions the <strong>standard error<\/strong> of [latex]\\hat{p}[\/latex]:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: center;\">Standard Error: [latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/span><\/p>\r\n<\/section>\r\n<p>Simulation provides one way to estimate the standard deviation of the sample proportion, and this formula gives another way.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1581[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1796[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use technology to create a sampling distribution of a sample proportion given [latex]n[\/latex] and [latex]p[\/latex].<\/li>\n<li>Calculate the mean and standard deviation for a sampling distribution of a sample proportion.<\/li>\n<li>Recognize the difference between the standard deviation and the standard error of a sample proportion.<\/li>\n<\/ul>\n<\/section>\n<h2>Standard Deviation vs. Standard Error<\/h2>\n<p>The previous exercise assumed that we knew the value of the population proportion and could either calculate the mean and standard deviation of the sample proportion by formulas, or estimate the mean and standard deviation by simulation. In practice, we do not typically know the population proportion, nor do we have the luxury of taking thousands of random samples. Instead, we observe a single random sample. In this case, we need to <strong>estimate<\/strong> the mean and standard deviation of the sample proportion.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>The estimated mean and standard deviation of the sampling distribution of a sample proportion<\/h3>\n<ul>\n<li>The <strong>estimated\u00a0<\/strong>mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\n<li>To distinguish it from the true standard deviation of sample proportions, we call the <strong>estimated<\/strong> standard deviation of sample proportions the <strong>standard error<\/strong> of [latex]\\hat{p}[\/latex]:<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: center;\">Standard Error: [latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/span><\/p>\n<\/section>\n<p>Simulation provides one way to estimate the standard deviation of the sample proportion, and this formula gives another way.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1581\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1581&theme=lumen&iframe_resize_id=ohm1581&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1796\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1796&theme=lumen&iframe_resize_id=ohm1796&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":17,"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\/1912"}],"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\/12"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1912\/revisions"}],"predecessor-version":[{"id":6708,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1912\/revisions\/6708"}],"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\/1912\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1912"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1912"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1912"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}