{"id":1343,"date":"2023-06-22T02:17:50","date_gmt":"2023-06-22T02:17:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-4\/"},"modified":"2025-05-16T22:42:23","modified_gmt":"2025-05-16T22:42:23","slug":"comparing-two-population-means-dependent-samples-apply-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-4\/","title":{"raw":"Comparing Two Population Means (Dependent Samples): Learn It 4","rendered":"Comparing Two Population Means (Dependent Samples): Learn It 4"},"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;Complete a two-sample t-test for dependent population means from hypotheses to conclusions&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;}\">Complete a two-sample [latex]t[\/latex]-test for dependent population means from hypotheses to conclusions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Hypothesis Testing for Dependent Samples (continued)<\/h2>\r\n<p>The third step in hypothesis testing is to calculate a test statistic, which we will utilize to find the P-value, write a conclusion, and make an inference about the population.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>test statistic ([latex]t[\/latex])<\/h3>\r\n<p>The notations for the summary statistics used to compare paired populations\/samples are shown in the following table. We will use [latex]d[\/latex]\u00a0to represent the difference variable.<\/p>\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 458.125px;\"><strong>Summary Statistics<\/strong><\/td>\r\n<td style=\"height: 18px; width: 235.208px;\"><strong>Notation<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 458.125px;\">Population Mean of Difference<\/td>\r\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\mu_d[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 458.125px;\">Sample Mean of Difference<\/td>\r\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\bar{d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 458.125px;\">Population Standard Deviation of Difference<\/td>\r\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\sigma_d[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 458.125px;\">Sample Standard Deviation of Difference<\/td>\r\n<td style=\"height: 18px; width: 235.208px;\">[latex]s_d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p class=\"para\">The <strong>test statistic for the dependent (paired) t-test<\/strong> is calculated using the following formulas:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{standard error of the difference}=\\dfrac{s_d}{\\sqrt{n}}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{test statistic }(t)=\\dfrac{\\text{estimator - null value}}{\\text{standard error of estimator}}=\\dfrac{\\bar{d}-\\text{null value}}{\\text{standard error of difference}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox interact\">Use the following steps:\r\n\r\n<p style=\"padding-left: 40px;\"><strong>Step 1: <\/strong>Click on the tab\u00a0<strong>Two Dependent Samples<\/strong>.<\/p>\r\n<p style=\"padding-left: 40px;\"><strong>Step 2: <\/strong>In the \u201cDataset\u201d drop-down menu, choose \u201cReaction Times (Paired Experiment).\u201d<\/p>\r\n<p style=\"padding-left: 40px;\"><strong>Step 3: <\/strong>In the left column, go to the drop-down menu for \u201cType of Inference\u201d and select \u201cSignificance Test.\u201d<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/2samplemean\/ \" width=\"100%\" height=\"1150\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2samplemean\/\" 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]1487[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">When thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the sample means, [latex]\\bar{x}_1-\\bar{x}_2[\/latex], is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_d[\/latex].\r\n\r\n<p>Take advantage of the statistical tool to calculate the standard deviation of the difference in the sample means.<\/p>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a two-sample t-test for dependent population means from hypotheses to conclusions&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;}\">Complete a two-sample [latex]t[\/latex]-test for dependent population means from hypotheses to conclusions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Hypothesis Testing for Dependent Samples (continued)<\/h2>\n<p>The third step in hypothesis testing is to calculate a test statistic, which we will utilize to find the P-value, write a conclusion, and make an inference about the population.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>test statistic ([latex]t[\/latex])<\/h3>\n<p>The notations for the summary statistics used to compare paired populations\/samples are shown in the following table. We will use [latex]d[\/latex]\u00a0to represent the difference variable.<\/p>\n<table>\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 458.125px;\"><strong>Summary Statistics<\/strong><\/td>\n<td style=\"height: 18px; width: 235.208px;\"><strong>Notation<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 458.125px;\">Population Mean of Difference<\/td>\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\mu_d[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 458.125px;\">Sample Mean of Difference<\/td>\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\bar{d}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 458.125px;\">Population Standard Deviation of Difference<\/td>\n<td style=\"height: 18px; width: 235.208px;\">[latex]\\sigma_d[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 458.125px;\">Sample Standard Deviation of Difference<\/td>\n<td style=\"height: 18px; width: 235.208px;\">[latex]s_d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p class=\"para\">The <strong>test statistic for the dependent (paired) t-test<\/strong> is calculated using the following formulas:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{standard error of the difference}=\\dfrac{s_d}{\\sqrt{n}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{test statistic }(t)=\\dfrac{\\text{estimator - null value}}{\\text{standard error of estimator}}=\\dfrac{\\bar{d}-\\text{null value}}{\\text{standard error of difference}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox interact\">Use the following steps:<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 1: <\/strong>Click on the tab\u00a0<strong>Two Dependent Samples<\/strong>.<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 2: <\/strong>In the \u201cDataset\u201d drop-down menu, choose \u201cReaction Times (Paired Experiment).\u201d<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 3: <\/strong>In the left column, go to the drop-down menu for \u201cType of Inference\u201d and select \u201cSignificance Test.\u201d<\/p>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/2samplemean\/\" width=\"100%\" height=\"1150\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2samplemean\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1487\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1487&theme=lumen&iframe_resize_id=ohm1487&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\">When thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the sample means, [latex]\\bar{x}_1-\\bar{x}_2[\/latex], is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_d[\/latex].<\/p>\n<p>Take advantage of the statistical tool to calculate the standard deviation of the difference in the sample means.<\/p>\n<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":29,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1309,"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\/1343"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1343\/revisions"}],"predecessor-version":[{"id":6825,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1343\/revisions\/6825"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1309"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1343\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1343"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1343"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1343"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}