{"id":1334,"date":"2023-06-22T02:17:43","date_gmt":"2023-06-22T02:17:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-apply-it-2\/"},"modified":"2025-05-16T22:33:59","modified_gmt":"2025-05-16T22:33:59","slug":"comparing-two-population-means-independent-samples-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-apply-it-2\/","title":{"raw":"Comparing Two Population Means (Independent Samples): Apply It 2","rendered":"Comparing Two Population Means (Independent Samples): Apply 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 independent 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 independent population means from hypotheses to conclusions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Test Statistics<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>[latex]t[\/latex] - statistic<\/h3>\r\n<p>The test statistic to compare two population means is calculated using the following formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]t = \\dfrac{\\text{estimate of parameter - null hypothesis value}}{\\text{standard error}} = \\dfrac{(\\bar{x}_1-\\bar{x}_2)-(\\mu_1-\\mu_2)}{\\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}}[\/latex]<\/p>\r\n<\/section>\r\n<p>The summary for the Hate Crime set can be seen below:<\/p>\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td><strong>Group 1<\/strong>: <i>Year 2019<\/i><\/td>\r\n<td><strong>Group 2<\/strong>: Year 2020<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sample Mean<\/strong><\/td>\r\n<td>[latex]\\bar{x}_1 =2.36[\/latex]<\/td>\r\n<td>[latex]\\bar{x}_2 =3.13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sample Standard Deviation<\/strong><\/td>\r\n<td>[latex]s_1 =1.79[\/latex]<\/td>\r\n<td>[latex]s_2 =2.04[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sample Size<\/strong><\/td>\r\n<td>[latex]n_1 =47[\/latex]<\/td>\r\n<td>[latex]n_2 =47[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p>Let's conduct the hypothesis test using the statistical tool below.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Under \"Enter Data\", select \"Summary Statistics\".<br \/>\r\n<strong>Step 2: <\/strong>Edit the Group Label accordingly<br \/>\r\n<strong>Step 3:<\/strong> Enter the statistics into the statistical tool<br \/>\r\n<strong>Step 4:<\/strong> Select \"Significance Test\" for the \"Type of Inference\"<\/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]5529[\/ohm2_question]<\/section>\r\n<p>We often learn more from constructing confidence intervals than from the hypothesis test because it shows a range of plausible values for the difference between the population means.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]5544[\/ohm2_question]<\/section>\r\n<section>Learn more about Hate Crime:<\/section>\r\n<ul>\r\n\t<li><a href=\"https:\/\/www.justice.gov\/hatecrimes\/learn-about-hate-crimes\">https:\/\/www.justice.gov\/hatecrimes\/learn-about-hate-crimes<\/a><\/li>\r\n\t<li><a href=\"https:\/\/www.whitehouse.gov\/briefing-room\/statements-releases\/2023\/04\/05\/icymi-president-joe-biden-op-ed-to-fight-antisemitism-we-must-remember-speak-out-and-act\/\">https:\/\/www.whitehouse.gov\/briefing-room\/statements-releases\/2023\/04\/05\/icymi-president-joe-biden-op-ed-to-fight-antisemitism-we-must-remember-speak-out-and-act\/<\/a><\/li>\r\n<\/ul>\r\n<section><\/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 independent 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 independent population means from hypotheses to conclusions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Test Statistics<\/h2>\n<section class=\"textbox keyTakeaway\">\n<h3>[latex]t[\/latex] &#8211; statistic<\/h3>\n<p>The test statistic to compare two population means is calculated using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]t = \\dfrac{\\text{estimate of parameter - null hypothesis value}}{\\text{standard error}} = \\dfrac{(\\bar{x}_1-\\bar{x}_2)-(\\mu_1-\\mu_2)}{\\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}}[\/latex]<\/p>\n<\/section>\n<p>The summary for the Hate Crime set can be seen below:<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td><strong>Group 1<\/strong>: <i>Year 2019<\/i><\/td>\n<td><strong>Group 2<\/strong>: Year 2020<\/td>\n<\/tr>\n<tr>\n<td><strong>Sample Mean<\/strong><\/td>\n<td>[latex]\\bar{x}_1 =2.36[\/latex]<\/td>\n<td>[latex]\\bar{x}_2 =3.13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Sample Standard Deviation<\/strong><\/td>\n<td>[latex]s_1 =1.79[\/latex]<\/td>\n<td>[latex]s_2 =2.04[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Sample Size<\/strong><\/td>\n<td>[latex]n_1 =47[\/latex]<\/td>\n<td>[latex]n_2 =47[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Let&#8217;s conduct the hypothesis test using the statistical tool below.<\/p>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Under &#8220;Enter Data&#8221;, select &#8220;Summary Statistics&#8221;.<br \/>\n<strong>Step 2: <\/strong>Edit the Group Label accordingly<br \/>\n<strong>Step 3:<\/strong> Enter the statistics into the statistical tool<br \/>\n<strong>Step 4:<\/strong> Select &#8220;Significance Test&#8221; for the &#8220;Type of Inference&#8221;<\/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=\"ohm5529\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5529&theme=lumen&iframe_resize_id=ohm5529&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We often learn more from constructing confidence intervals than from the hypothesis test because it shows a range of plausible values for the difference between the population means.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm5544\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5544&theme=lumen&iframe_resize_id=ohm5544&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>Learn more about Hate Crime:<\/section>\n<ul>\n<li><a href=\"https:\/\/www.justice.gov\/hatecrimes\/learn-about-hate-crimes\">https:\/\/www.justice.gov\/hatecrimes\/learn-about-hate-crimes<\/a><\/li>\n<li><a href=\"https:\/\/www.whitehouse.gov\/briefing-room\/statements-releases\/2023\/04\/05\/icymi-president-joe-biden-op-ed-to-fight-antisemitism-we-must-remember-speak-out-and-act\/\">https:\/\/www.whitehouse.gov\/briefing-room\/statements-releases\/2023\/04\/05\/icymi-president-joe-biden-op-ed-to-fight-antisemitism-we-must-remember-speak-out-and-act\/<\/a><\/li>\n<\/ul>\n<section><\/section>\n","protected":false},"author":8,"menu_order":24,"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":"apply_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\/1334"}],"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\/1334\/revisions"}],"predecessor-version":[{"id":6820,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1334\/revisions\/6820"}],"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\/1334\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1334"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1334"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1334"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}