{"id":1986,"date":"2023-07-22T00:29:33","date_gmt":"2023-07-22T00:29:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=1986"},"modified":"2025-05-16T03:18:25","modified_gmt":"2025-05-16T03:18:25","slug":"confidence-intervals-for-the-difference-in-population-proportions-learn-it-3-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-intervals-for-the-difference-in-population-proportions-learn-it-3-2\/","title":{"raw":"Confidence Intervals for the Difference in Population Proportions: Learn It 3","rendered":"Confidence Intervals for the Difference in Population Proportions: 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;Calculate a confidence interval for the difference in proportions of two groups&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;}\">Calculate a confidence interval for the difference in proportions of two groups.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Make conclusions based on a confidence interval&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;}\">Make conclusions based on a confidence interval.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Difference in Proportions<\/h2>\r\n<p>Our primary goal is to use the data to examine whether there\u2019s a difference in the proportion of callbacks for applications the researchers identified as being perceived as female and the proportion of callbacks for applications the researchers identified as being perceived as male.<\/p>\r\n<p>It is not feasible to simulate every possible sample to derive an exact sampling distribution in this scenario. Instead, we can use mathematical theory to derive expressions for the mean and standard deviation of the sampling distribution for the difference in proportions.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Sampling Distribution for the Difference in Proportions<\/h3>\r\n<p>When certain conditions apply, the sampling distribution tells us three things about the distribution of [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex]:<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>For large samples, the distribution is normal.<\/li>\r\n\t<li>The distribution has a <strong>mean<\/strong> of [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex].<\/li>\r\n\t<li>The distribution has a <strong>standard deviation<\/strong> of<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center;\">[latex] \\sqrt{\\frac{p_{1}(1-p_{1})}{n_{1}} + \\frac{p_{2}(1-p_{2})}{n_{2}}}[\/latex].<\/p>\r\n<p style=\"padding-left: 40px; text-align: left;\">Similar to previous calculations, we will replace [latex] p_{1} [\/latex] and [latex] p_{2} [\/latex] in the formula with the respective sample proportions [latex] \\hat{p}_{1} [\/latex] and [latex]\\hat{p}_{2}[\/latex]. The estimate is called the <strong>standard error<\/strong>. This is the estimate of the sample-to-sample variability, the random variability we expect in [latex] \\hat{p}_{1} - \\hat{p}_{2}[\/latex] if we take random samples of the same size repeatedly.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]11070[\/ohm2_question]<\/section>\r\n<p>Now that we have our estimate of the difference in proportions and the standard error, let\u2019s calculate the confidence interval.<\/p>\r\n<section class=\"textbox example\">\r\n<p style=\"text-align: center;\">Estimate\u00a0[latex] \\pm [\/latex] Margin of Error<\/p>\r\n<p style=\"text-align: center;\">[latex] (\\hat{p}_1 - \\hat{p}_2) \\pm z^{*} \\times \\sqrt{\\frac{\\hat{p}_1(1-\\hat{p}_1)}{n_{1}} + \\frac{\\hat{p}_2(1-\\hat{p}_2)}{n_{2}}} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] 0.00864 \\pm 1.96^{*} \\times 0.009 [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] 0.00864 \\pm 0.01764 [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] 0.00864 - 0.01764 = -0.009 [\/latex] and [latex] 0.00864 + 0.01764 = .026 [\/latex]<\/p>\r\n<p>This gives us a confidence interval of [latex](-0.009,.026)[\/latex] for the true difference in the proportion of applicants with a female-perceived name and a male-perceived name. Notice how [latex]0[\/latex] is in the interval? This tells us that there is reasonably no significant difference in the true proportions.<\/p>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p>The [latex]z[\/latex]* critical value for [latex]90%[\/latex], [latex]95%[\/latex], and [latex]99%[\/latex] are [latex]1.645[\/latex], [latex]1.96[\/latex], and [latex]2.576[\/latex] respectively.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Confidence Interval for Difference in Proportions<\/h3>\r\n<p style=\"text-align: center;\">Estimate\u00a0[latex] \\pm [\/latex] Margin of Error<\/p>\r\n<p style=\"text-align: center;\">[latex] (\\hat{p}_1 - \\hat{p}_2) \\pm z^{*} \\times \\sqrt{\\frac{\\hat{p}_1(1-\\hat{p}_1)}{n_{1}} + \\frac{\\hat{p}_2(1-\\hat{p}_2)}{n_{2}}} [\/latex]<\/p>\r\n<ul>\r\n\t<li>The estimate is the difference in the sample proportions.<\/li>\r\n\t<li>The margin of error is the width of the confidence interval and is comprised to two parts:\r\n\r\n<ul>\r\n\t<li>[latex] z^{*}[\/latex]: The z critical value; this is the point on the standard normal distribution such that the proportion of area under the curve between\u00a0[latex] -z^{*}[\/latex] and\u00a0[latex] +z^{*}[\/latex] is [latex] C[\/latex], the confidence level.<\/li>\r\n\t<li>Standard error: A measure of the sample-to-sample variability.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>In practice, we can either use the formula or use technology to calculate the confidence interval. Let's use technology to calculate the confidence interval.<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval for the difference in proportions of two groups&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;}\">Calculate a confidence interval for the difference in proportions of two groups.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Make conclusions based on a confidence interval&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;}\">Make conclusions based on a confidence interval.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Difference in Proportions<\/h2>\n<p>Our primary goal is to use the data to examine whether there\u2019s a difference in the proportion of callbacks for applications the researchers identified as being perceived as female and the proportion of callbacks for applications the researchers identified as being perceived as male.<\/p>\n<p>It is not feasible to simulate every possible sample to derive an exact sampling distribution in this scenario. Instead, we can use mathematical theory to derive expressions for the mean and standard deviation of the sampling distribution for the difference in proportions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Sampling Distribution for the Difference in Proportions<\/h3>\n<p>When certain conditions apply, the sampling distribution tells us three things about the distribution of [latex]\\hat{p}_{1} - \\hat{p}_{2}[\/latex]:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>For large samples, the distribution is normal.<\/li>\n<li>The distribution has a <strong>mean<\/strong> of [latex]\\hat{p}_{1} - \\hat{p}_{2}[\/latex].<\/li>\n<li>The distribution has a <strong>standard deviation<\/strong> of<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]\\sqrt{\\frac{p_{1}(1-p_{1})}{n_{1}} + \\frac{p_{2}(1-p_{2})}{n_{2}}}[\/latex].<\/p>\n<p style=\"padding-left: 40px; text-align: left;\">Similar to previous calculations, we will replace [latex]p_{1}[\/latex] and [latex]p_{2}[\/latex] in the formula with the respective sample proportions [latex]\\hat{p}_{1}[\/latex] and [latex]\\hat{p}_{2}[\/latex]. The estimate is called the <strong>standard error<\/strong>. This is the estimate of the sample-to-sample variability, the random variability we expect in [latex]\\hat{p}_{1} - \\hat{p}_{2}[\/latex] if we take random samples of the same size repeatedly.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm11070\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=11070&theme=lumen&iframe_resize_id=ohm11070&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Now that we have our estimate of the difference in proportions and the standard error, let\u2019s calculate the confidence interval.<\/p>\n<section class=\"textbox example\">\n<p style=\"text-align: center;\">Estimate\u00a0[latex]\\pm[\/latex] Margin of Error<\/p>\n<p style=\"text-align: center;\">[latex](\\hat{p}_1 - \\hat{p}_2) \\pm z^{*} \\times \\sqrt{\\frac{\\hat{p}_1(1-\\hat{p}_1)}{n_{1}} + \\frac{\\hat{p}_2(1-\\hat{p}_2)}{n_{2}}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0.00864 \\pm 1.96^{*} \\times 0.009[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0.00864 \\pm 0.01764[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0.00864 - 0.01764 = -0.009[\/latex] and [latex]0.00864 + 0.01764 = .026[\/latex]<\/p>\n<p>This gives us a confidence interval of [latex](-0.009,.026)[\/latex] for the true difference in the proportion of applicants with a female-perceived name and a male-perceived name. Notice how [latex]0[\/latex] is in the interval? This tells us that there is reasonably no significant difference in the true proportions.<\/p>\n<\/section>\n<section class=\"textbox recall\">\n<p>The [latex]z[\/latex]* critical value for [latex]90%[\/latex], [latex]95%[\/latex], and [latex]99%[\/latex] are [latex]1.645[\/latex], [latex]1.96[\/latex], and [latex]2.576[\/latex] respectively.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>Confidence Interval for Difference in Proportions<\/h3>\n<p style=\"text-align: center;\">Estimate\u00a0[latex]\\pm[\/latex] Margin of Error<\/p>\n<p style=\"text-align: center;\">[latex](\\hat{p}_1 - \\hat{p}_2) \\pm z^{*} \\times \\sqrt{\\frac{\\hat{p}_1(1-\\hat{p}_1)}{n_{1}} + \\frac{\\hat{p}_2(1-\\hat{p}_2)}{n_{2}}}[\/latex]<\/p>\n<ul>\n<li>The estimate is the difference in the sample proportions.<\/li>\n<li>The margin of error is the width of the confidence interval and is comprised to two parts:\n<ul>\n<li>[latex]z^{*}[\/latex]: The z critical value; this is the point on the standard normal distribution such that the proportion of area under the curve between\u00a0[latex]-z^{*}[\/latex] and\u00a0[latex]+z^{*}[\/latex] is [latex]C[\/latex], the confidence level.<\/li>\n<li>Standard error: A measure of the sample-to-sample variability.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<p>In practice, we can either use the formula or use technology to calculate the confidence interval. Let&#8217;s use technology to calculate the confidence interval.<\/p>\n","protected":false},"author":12,"menu_order":30,"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\/1986"}],"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":17,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1986\/revisions"}],"predecessor-version":[{"id":6740,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1986\/revisions\/6740"}],"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\/1986\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1986"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1986"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1986"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}