{"id":1302,"date":"2023-06-22T02:13:31","date_gmt":"2023-06-22T02:13:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-learn-it-3\/"},"modified":"2025-05-16T04:05:30","modified_gmt":"2025-05-16T04:05:30","slug":"confidence-interval-for-difference-in-population-means-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-learn-it-3\/","title":{"raw":"Confidence Interval for Difference in Population Means: Learn It 3","rendered":"Confidence Interval for Difference in Population Means: Learn It 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/li>\r\n\t<li class=\"li1\">Calculate and explain a confidence interval for the difference between two population means.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Confidence Interval for Difference in Population Means<\/h2>\r\n<p>In previous lessons, you constructed confidence interval estimates for a population proportion, a difference in proportions, and a population mean.<\/p>\r\n<p>The form of those confidence intervals was:<\/p>\r\n<p style=\"text-align: center;\"><strong>estimate [latex]\\pm [\/latex] margin of error<\/strong><\/p>\r\n<p>When you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\\bar{x}_1 - \\bar{x}_2[\/latex], and the margin of error is calculated using the standard error for a difference in sample means and a critical value from the [latex]t[\/latex]-distribution.<\/p>\r\n<p>Compared to the formula for proportions, the margin of error here is calculated a little differently: Instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate [latex]t[\/latex]-distribution. This is not surprising if you think back to your work with the standardized [latex]t[\/latex]-statistic.<\/p>\r\n<p style=\"text-align: center;\"><strong>Margin of Error<\/strong> = ([latex]t[\/latex]-critical)(standard error) = ([latex]t[\/latex]-critical)[latex](\\sqrt{\\frac{{{s}_{1}}^{2}}{{n}_{1}}+\\frac{{{s}_{2}}^{2}}{{n}_{2}}})[\/latex]<\/p>\r\n<p>We will use technology to calculate the margin of error, so we won\u2019t worry about remembering the formula for standard error or margin of error at this point.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1919[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/li>\n<li class=\"li1\">Calculate and explain a confidence interval for the difference between two population means.<\/li>\n<\/ul>\n<\/section>\n<h2>Confidence Interval for Difference in Population Means<\/h2>\n<p>In previous lessons, you constructed confidence interval estimates for a population proportion, a difference in proportions, and a population mean.<\/p>\n<p>The form of those confidence intervals was:<\/p>\n<p style=\"text-align: center;\"><strong>estimate [latex]\\pm[\/latex] margin of error<\/strong><\/p>\n<p>When you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\\bar{x}_1 - \\bar{x}_2[\/latex], and the margin of error is calculated using the standard error for a difference in sample means and a critical value from the [latex]t[\/latex]-distribution.<\/p>\n<p>Compared to the formula for proportions, the margin of error here is calculated a little differently: Instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate [latex]t[\/latex]-distribution. This is not surprising if you think back to your work with the standardized [latex]t[\/latex]-statistic.<\/p>\n<p style=\"text-align: center;\"><strong>Margin of Error<\/strong> = ([latex]t[\/latex]-critical)(standard error) = ([latex]t[\/latex]-critical)[latex](\\sqrt{\\frac{{{s}_{1}}^{2}}{{n}_{1}}+\\frac{{{s}_{2}}^{2}}{{n}_{2}}})[\/latex]<\/p>\n<p>We will use technology to calculate the margin of error, so we won\u2019t worry about remembering the formula for standard error or margin of error at this point.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1919\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1919&theme=lumen&iframe_resize_id=ohm1919&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"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":1268,"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\/1302"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1302\/revisions"}],"predecessor-version":[{"id":6798,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1302\/revisions\/6798"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1268"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1302\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1302"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1302"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1302"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1302"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}