{"id":1342,"date":"2023-06-22T02:17:49","date_gmt":"2023-06-22T02:17:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-3\/"},"modified":"2025-05-16T22:41:59","modified_gmt":"2025-05-16T22:41:59","slug":"comparing-two-population-means-dependent-samples-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-3\/","title":{"raw":"Comparing Two Population Means (Dependent Samples): Learn It 3","rendered":"Comparing Two Population Means (Dependent Samples): 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;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<\/h2>\r\n<p>Let's conduct the hypothesis test for the difference in drivers' reaction times when they are using a cell phone as opposed to when they are not using a cell phone.<\/p>\r\n<p>[reveal-answer q=\"630785\"]<strong>Steps for Hypothesis Testing<\/strong>[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"630785\"]<\/p>\r\n<ol>\r\n\t<li>Write out the null and alternative hypotheses.<\/li>\r\n\t<li>Check the conditions for the hypothesis test.<\/li>\r\n\t<li>Calculate a test statistic.<\/li>\r\n\t<li>Calculate a P-value.<\/li>\r\n\t<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\r\n\t<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>null and alternative hypotheses<\/h3>\r\n<p>A <strong>dependent or<\/strong> <strong>paired [latex]t[\/latex]-test<\/strong> compares the mean of the differences, [latex]\\mu_d[\/latex], to a hypothesized value, which is often [latex]0[\/latex], but not always. Thus, a dependent [latex]t[\/latex]-test is the same as a one-sample [latex]t[\/latex]-test performed on the difference variable, [latex]d[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>In summary, where [latex]k[\/latex]\u00a0is the value of the null hypothesis, we have:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Null Hypothesis for <\/strong><strong>Independent Samples<\/strong><\/td>\r\n<td style=\"text-align: center;\"><strong>Null Hypothesis for <\/strong><strong>Dependent Samples<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]H_0: \\mu_1-\\mu_2=k[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]H_0: \\mu_d=k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\"><strong>Alternative Hypothesis <\/strong><strong>for Independent Samples<\/strong><\/td>\r\n<td style=\"text-align: center;\"><strong>Alternative Hypothesis <\/strong><strong>for Dependent Samples<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2&gt;k[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d&gt;k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2 \\lt k[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d \\lt k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2 \\ne k[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d \\ne k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1973[\/ohm2_question]<\/section>\r\n<p>It is always important to check the assumptions of a test before you perform any calculations.<\/p>\r\n<p>[reveal-answer q=\"175098\"]<strong>Conditions for a One-Sample [latex]t[\/latex]-Test<\/strong> [\/reveal-answer]<br \/>\r\n[hidden-answer a=\"175098\"]<\/p>\r\n<ol>\r\n\t<li>The sample is a\u00a0<strong>random sample<\/strong> from the population of interest, or it is reasonable to regard the sample as random. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in a sample that is representative of the population.<\/li>\r\n\t<li>For each population, the distribution of the variable that was measured is\u00a0<strong>approximately normal, or the sample size for the sample from that population is large<\/strong>. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]894[\/ohm2_question]<\/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<\/h2>\n<p>Let&#8217;s conduct the hypothesis test for the difference in drivers&#8217; reaction times when they are using a cell phone as opposed to when they are not using a cell phone.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q630785\"><strong>Steps for Hypothesis Testing<\/strong><\/button><\/p>\n<div id=\"q630785\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Write out the null and alternative hypotheses.<\/li>\n<li>Check the conditions for the hypothesis test.<\/li>\n<li>Calculate a test statistic.<\/li>\n<li>Calculate a P-value.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\n<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3>null and alternative hypotheses<\/h3>\n<p>A <strong>dependent or<\/strong> <strong>paired [latex]t[\/latex]-test<\/strong> compares the mean of the differences, [latex]\\mu_d[\/latex], to a hypothesized value, which is often [latex]0[\/latex], but not always. Thus, a dependent [latex]t[\/latex]-test is the same as a one-sample [latex]t[\/latex]-test performed on the difference variable, [latex]d[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>In summary, where [latex]k[\/latex]\u00a0is the value of the null hypothesis, we have:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Null Hypothesis for <\/strong><strong>Independent Samples<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Null Hypothesis for <\/strong><strong>Dependent Samples<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]H_0: \\mu_1-\\mu_2=k[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]H_0: \\mu_d=k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Alternative Hypothesis <\/strong><strong>for Independent Samples<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Alternative Hypothesis <\/strong><strong>for Dependent Samples<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2>k[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d>k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2 \\lt k[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d \\lt k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_1-\\mu_2 \\ne k[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]H_A: \\mu_d \\ne k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1973\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1973&theme=lumen&iframe_resize_id=ohm1973&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>It is always important to check the assumptions of a test before you perform any calculations.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q175098\"><strong>Conditions for a One-Sample [latex]t[\/latex]-Test<\/strong> <\/button><\/p>\n<div id=\"q175098\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The sample is a\u00a0<strong>random sample<\/strong> from the population of interest, or it is reasonable to regard the sample as random. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in a sample that is representative of the population.<\/li>\n<li>For each population, the distribution of the variable that was measured is\u00a0<strong>approximately normal, or the sample size for the sample from that population is large<\/strong>. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm894\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=894&theme=lumen&iframe_resize_id=ohm894&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":28,"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\/1342"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1342\/revisions"}],"predecessor-version":[{"id":6824,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1342\/revisions\/6824"}],"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\/1342\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1342"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1342"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1342"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}