{"id":1341,"date":"2023-06-22T02:17:48","date_gmt":"2023-06-22T02:17:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-2\/"},"modified":"2025-05-16T22:40:55","modified_gmt":"2025-05-16T22:40:55","slug":"comparing-two-population-means-dependent-samples-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-dependent-samples-apply-it-2\/","title":{"raw":"Comparing Two Population Means (Dependent Samples): Learn It 2","rendered":"Comparing Two Population Means (Dependent Samples): Learn It 2"},"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>Differences in Reaction Times<\/h2>\r\n<p>In a hypothesis test for matched or paired samples, subjects are matched in pairs, and differences are calculated. Let's calculate the point estimate of the mean of differences between two paired samples. That is, let's calculate the difference between the reaction time of each participant with the use of a cell phone and without the use of a cell phone.<\/p>\r\n<p>The difference is defined by:<\/p>\r\n<p>Difference = (reaction time with the use of a cell phone) \u2013 (reaction time without the use of a cell phone)<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1484[\/ohm2_question]<\/section>\r\n<p>Let's calculate the sample mean difference and the sample standard deviation of the differences using our statistical tool.<\/p>\r\n<section class=\"textbox interact\">Under \"Enter Data\", select \"Your Own\". Copy and paste the following difference data.<br \/>\r\n[reveal-answer q=\"216189\"]Difference Column [\/reveal-answer]<br \/>\r\n[hidden-answer a=\"216189\"]\r\n\r\n<table style=\"height: 1203px; width: 56px;\" width=\"102\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 229.861px;\">Difference<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">32<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">67<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">75<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">150<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">142<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">29<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">84<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">22<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">29<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">19<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">87<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">101<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">45<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">66<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">73<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">148<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">130<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">90<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">146<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">39<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">20<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 229.861px;\">-7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/ \" width=\"100%\" height=\"900\" 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\/eda_quantitative\/\" 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]1485[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">The mean of the differences is equal to the difference in means.\r\n\r\n<p style=\"text-align: center;\">[latex]\\mu_d = \\mu_\\text{with cell phone}-\\mu_\\text{without cell phone}[\/latex]<\/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>Differences in Reaction Times<\/h2>\n<p>In a hypothesis test for matched or paired samples, subjects are matched in pairs, and differences are calculated. Let&#8217;s calculate the point estimate of the mean of differences between two paired samples. That is, let&#8217;s calculate the difference between the reaction time of each participant with the use of a cell phone and without the use of a cell phone.<\/p>\n<p>The difference is defined by:<\/p>\n<p>Difference = (reaction time with the use of a cell phone) \u2013 (reaction time without the use of a cell phone)<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1484\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1484&theme=lumen&iframe_resize_id=ohm1484&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Let&#8217;s calculate the sample mean difference and the sample standard deviation of the differences using our statistical tool.<\/p>\n<section class=\"textbox interact\">Under &#8220;Enter Data&#8221;, select &#8220;Your Own&#8221;. Copy and paste the following difference data.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q216189\">Difference Column <\/button><\/p>\n<div id=\"q216189\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"height: 1203px; width: 56px; width: 102px;\">\n<tbody>\n<tr>\n<td style=\"width: 229.861px;\">Difference<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">32<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">67<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">75<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">150<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">142<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">56<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">84<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-13<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-11<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">22<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-24<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">19<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">87<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">101<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">45<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">66<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">73<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">148<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">130<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">90<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">146<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">39<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">20<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 229.861px;\">-7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"900\" 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\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1485\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1485&theme=lumen&iframe_resize_id=ohm1485&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\">The mean of the differences is equal to the difference in means.<\/p>\n<p style=\"text-align: center;\">[latex]\\mu_d = \\mu_\\text{with cell phone}-\\mu_\\text{without cell phone}[\/latex]<\/p>\n<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":27,"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\/1341"}],"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\/1341\/revisions"}],"predecessor-version":[{"id":6823,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1341\/revisions\/6823"}],"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\/1341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}