{"id":1168,"date":"2023-06-22T01:59:36","date_gmt":"2023-06-22T01:59:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-learn-it-2\/"},"modified":"2025-05-16T03:06:59","modified_gmt":"2025-05-16T03:06:59","slug":"confidence-interval-for-proportions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-learn-it-2\/","title":{"raw":"Confidence Interval for Proportions: Learn It 2","rendered":"Confidence Interval for Proportions: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Check the conditions for creating a confidence interval for population proportion.<\/li>\r\n\t<li>Describe the connection between the confidence level and the confidence interval.<\/li>\r\n\t<li>Calculate a confidence interval for a population proportion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox recall\"><strong><strong><strong>Sampling a distribution of the sample proportion<\/strong><\/strong><\/strong><br \/>\r\n<br \/>\r\nWhen taking random samples of size [latex]n[\/latex] from a population distribution with proportion [latex]p[\/latex]:\r\n\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>If [latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the <strong>Central Limit Theorem<\/strong> (CLT) states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[\/latex] 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]1627[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1037[\/ohm2_question]<\/section>\r\n<h3>Standard error<\/h3>\r\n<p>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]. This is called the <strong>standard error<\/strong>, the estimated standard deviation of sample proportions. It is the measure of sample-to-sample variability. We will use the standard error to help us convey information about the accuracy of our point estimate.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1629[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Check the conditions for creating a confidence interval for population proportion.<\/li>\n<li>Describe the connection between the confidence level and the confidence interval.<\/li>\n<li>Calculate a confidence interval for a population proportion.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\"><strong><strong><strong>Sampling a distribution of the sample proportion<\/strong><\/strong><\/strong><\/p>\n<p>When taking 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>If [latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex], then the <strong>Central Limit Theorem<\/strong> (CLT) states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[\/latex] 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=\"ohm1627\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1627&theme=lumen&iframe_resize_id=ohm1627&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1037\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1037&theme=lumen&iframe_resize_id=ohm1037&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Standard error<\/h3>\n<p>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]. This is called the <strong>standard error<\/strong>, the estimated standard deviation of sample proportions. It is the measure of sample-to-sample variability. We will use the standard error to help us convey information about the accuracy of our point estimate.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1629\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1629&theme=lumen&iframe_resize_id=ohm1629&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1163,"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\/1168"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1168\/revisions"}],"predecessor-version":[{"id":6721,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1168\/revisions\/6721"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1163"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1168\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1168"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1168"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1168"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}