{"id":1171,"date":"2023-06-22T01:59:38","date_gmt":"2023-06-22T01:59:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-learn-it-4\/"},"modified":"2025-05-16T03:07:56","modified_gmt":"2025-05-16T03:07:56","slug":"confidence-interval-for-proportions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-proportions-learn-it-4\/","title":{"raw":"Confidence Interval for Proportions: Learn It 4","rendered":"Confidence Interval for Proportions: Learn It 4"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Check the conditions for creating a confidence interval for population proportion.<\/li>\r\n\t<li class=\"li1\">Describe the connection between the confidence level and the confidence interval.<\/li>\r\n\t<li class=\"li1\">Calculate a confidence interval for a population proportion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>The [latex]z[\/latex] critical value, [latex]z^{*}[\/latex]<\/h3>\r\n<p>Let's find [latex]z^{*}[\/latex] value using the Normal Distribution Statistical Tool.<\/p>\r\n<section class=\"textbox example\">Find the [latex]z[\/latex] critical value for a [latex]95\\%[\/latex] confidence level. On the statistical tool below:\r\n\r\n<ul>\r\n\t<li>Select the <strong>Find Percentile\/Quantile<\/strong>\u00a0tab.<\/li>\r\n\t<li>The tool defaults to a standard normal distribution with mean [latex] \\mu = 0 [\/latex]and standard deviation [latex] \\sigma = 1[\/latex].<\/li>\r\n\t<li>Select <strong>Two-tailed<\/strong> and enter [latex]95[\/latex] for <strong>Central Probability<\/strong>, since we want the middle [latex]95\\%[\/latex].<\/li>\r\n<\/ul>\r\n<p>The [latex]z[\/latex] critical value, [latex]z^{*}[\/latex], for a [latex] 95\\%[\/latex] confidence level is <strong>[latex]1.96[\/latex]<\/strong>.<\/p>\r\n<p>Note the tool presents both the [latex]z[\/latex] critical value and its negative counterpart, [latex]-1.96[\/latex].<\/p>\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]1628[\/ohm2_question]<\/section>\r\n<h3>Margin of error<\/h3>\r\n<p>Now that we have the standard error and [latex]z[\/latex] critical value, we can calculate the margin of error that we can use to find the confidence interval for a population proportion.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>margin of error, [latex]ME[\/latex]<\/h3>\r\n<p><strong>Margin of error<\/strong> is the width of the confidence interval.<\/p>\r\n<p style=\"text-align: center;\">[latex]ME = z^{*} \\cdot (\\text{standard error})[\/latex]<\/p>\r\n<p>where:<\/p>\r\n<ul>\r\n\t<li>standard error = [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}} [\/latex]<\/li>\r\n\t<li>[latex]z^{*}[\/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex]\u2212z^{*}[\/latex] and [latex]+z^{*}[\/latex] is [latex] C[\/latex], the confidence level.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1630[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Check the conditions for creating a confidence interval for population proportion.<\/li>\n<li class=\"li1\">Describe the connection between the confidence level and the confidence interval.<\/li>\n<li class=\"li1\">Calculate a confidence interval for a population proportion.<\/li>\n<\/ul>\n<\/section>\n<h3>The [latex]z[\/latex] critical value, [latex]z^{*}[\/latex]<\/h3>\n<p>Let&#8217;s find [latex]z^{*}[\/latex] value using the Normal Distribution Statistical Tool.<\/p>\n<section class=\"textbox example\">Find the [latex]z[\/latex] critical value for a [latex]95\\%[\/latex] confidence level. On the statistical tool below:<\/p>\n<ul>\n<li>Select the <strong>Find Percentile\/Quantile<\/strong>\u00a0tab.<\/li>\n<li>The tool defaults to a standard normal distribution with mean [latex]\\mu = 0[\/latex]and standard deviation [latex]\\sigma = 1[\/latex].<\/li>\n<li>Select <strong>Two-tailed<\/strong> and enter [latex]95[\/latex] for <strong>Central Probability<\/strong>, since we want the middle [latex]95\\%[\/latex].<\/li>\n<\/ul>\n<p>The [latex]z[\/latex] critical value, [latex]z^{*}[\/latex], for a [latex]95\\%[\/latex] confidence level is <strong>[latex]1.96[\/latex]<\/strong>.<\/p>\n<p>Note the tool presents both the [latex]z[\/latex] critical value and its negative counterpart, [latex]-1.96[\/latex].<\/p>\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=\"ohm1628\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1628&theme=lumen&iframe_resize_id=ohm1628&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Margin of error<\/h3>\n<p>Now that we have the standard error and [latex]z[\/latex] critical value, we can calculate the margin of error that we can use to find the confidence interval for a population proportion.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>margin of error, [latex]ME[\/latex]<\/h3>\n<p><strong>Margin of error<\/strong> is the width of the confidence interval.<\/p>\n<p style=\"text-align: center;\">[latex]ME = z^{*} \\cdot (\\text{standard error})[\/latex]<\/p>\n<p>where:<\/p>\n<ul>\n<li>standard error = [latex]\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/li>\n<li>[latex]z^{*}[\/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex]\u2212z^{*}[\/latex] and [latex]+z^{*}[\/latex] is [latex]C[\/latex], the confidence level.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1630\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1630&theme=lumen&iframe_resize_id=ohm1630&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":10,"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\/1171"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1171\/revisions"}],"predecessor-version":[{"id":6722,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1171\/revisions\/6722"}],"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\/1171\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1171"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1171"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1171"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}