{"id":1335,"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-dig-deeper-2\/"},"modified":"2025-05-16T22:36:58","modified_gmt":"2025-05-16T22:36:58","slug":"comparing-two-population-means-independent-samples-dig-deeper-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-dig-deeper-2\/","title":{"raw":"Comparing Two Population Means (Independent Samples): Fresh Take","rendered":"Comparing Two Population Means (Independent Samples): Fresh Take"},"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<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 example\">A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. Both populations have a normal distribution.\r\n\r\n<ul>\r\n\t<li>College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes.<\/li>\r\n\t<li>College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class.<\/li>\r\n<\/ul>\r\n<p>The community group believes that a student who graduates from College A\u00a0<strong>has taken more math classes<\/strong>, on average. Is the community group correct?<\/p>\r\n<p>To prove or disprove the community group's beliefs, we need to conduct a hypothesis test or utilize a confidence interval to make an inference.<\/p>\r\n<ul>\r\n\t<li>Using a hypothesis test, does the data suggest that College A has taken more math classes than College B?<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"508631\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"508631\"]<\/p>\r\n<p><strong>Step 1<\/strong>: Write out the null and alternative hypotheses.<\/p>\r\n<p style=\"padding-left: 40px;\">[latex]H_0: \\mu_A - \\mu_B = 0[\/latex]<\/p>\r\n<p style=\"padding-left: 40px;\">[latex]H_A: \\mu_A - \\mu_B \\gt 0[\/latex]<\/p>\r\n<p><strong>Step 2:\u00a0<\/strong>Check the conditions for the hypothesis test.<\/p>\r\n<p style=\"padding-left: 40px;\">Because the populations have a normal distribution, the conditions are met.<\/p>\r\n<p><strong>Step 3:\u00a0<\/strong>Calculate the [latex]t[\/latex]-test statistic.<\/p>\r\n<p style=\"padding-left: 40px;\">Using the statistical tool above, [latex]t=0.890[\/latex].<\/p>\r\n<p><strong>Step 4:\u00a0<\/strong>Calculate a P-value.<\/p>\r\n<p style=\"padding-left: 40px;\">Using the statistical tool above, P-value [latex]=0.1928[\/latex]. Note that we are conducting a one-tailed test because our alternative hypothesis is testing \"greater than\".<\/p>\r\n<p><strong>Step 5:\u00a0<\/strong>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/p>\r\n<p style=\"padding-left: 40px;\">The P-value [latex]=0.1928 &gt; \\alpha = 0.05[\/latex]. Therefore, there is not enough evidence to reject the null hypothesis.<\/p>\r\n<p><strong>Step 6:\u00a0<\/strong>Write a conclusion in context.<\/p>\r\n<p style=\"padding-left: 40px;\">At the [latex]5\\%[\/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.<\/p>\r\n<p>Therefore, the community college's beliefs are not correct.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<ul>\r\n\t<li>Using a confidence interval, does the data suggest that College A has taken more math classes than College B?<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"414734\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"414734\"]<\/p>\r\n<p>Using the statistical tool, we find that the [latex]95\\%[\/latex] confidence interval for this data set is [latex](-0.683, 1.683)[\/latex].<\/p>\r\n<p>Interpretation: We are [latex]95\\%[\/latex] confident that the true difference in the mean math classes between College A and College B is [latex]-0.683[\/latex] and [latex]1.683[\/latex] classes.<\/p>\r\n<p>The values in the [latex]95\\%[\/latex] confidence interval include some negative values, some positive values, and [latex]0[\/latex]. This implies that we cannot conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.<\/p>\r\n<p>We achieved the same conclusion as the hypothesis test.<\/p>\r\n<p>[\/hidden-answer]<\/p>\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 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<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 example\">A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. Both populations have a normal distribution.<\/p>\n<ul>\n<li>College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes.<\/li>\n<li>College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class.<\/li>\n<\/ul>\n<p>The community group believes that a student who graduates from College A\u00a0<strong>has taken more math classes<\/strong>, on average. Is the community group correct?<\/p>\n<p>To prove or disprove the community group&#8217;s beliefs, we need to conduct a hypothesis test or utilize a confidence interval to make an inference.<\/p>\n<ul>\n<li>Using a hypothesis test, does the data suggest that College A has taken more math classes than College B?<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q508631\">Show Answer<\/button><\/p>\n<div id=\"q508631\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1<\/strong>: Write out the null and alternative hypotheses.<\/p>\n<p style=\"padding-left: 40px;\">[latex]H_0: \\mu_A - \\mu_B = 0[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">[latex]H_A: \\mu_A - \\mu_B \\gt 0[\/latex]<\/p>\n<p><strong>Step 2:\u00a0<\/strong>Check the conditions for the hypothesis test.<\/p>\n<p style=\"padding-left: 40px;\">Because the populations have a normal distribution, the conditions are met.<\/p>\n<p><strong>Step 3:\u00a0<\/strong>Calculate the [latex]t[\/latex]-test statistic.<\/p>\n<p style=\"padding-left: 40px;\">Using the statistical tool above, [latex]t=0.890[\/latex].<\/p>\n<p><strong>Step 4:\u00a0<\/strong>Calculate a P-value.<\/p>\n<p style=\"padding-left: 40px;\">Using the statistical tool above, P-value [latex]=0.1928[\/latex]. Note that we are conducting a one-tailed test because our alternative hypothesis is testing &#8220;greater than&#8221;.<\/p>\n<p><strong>Step 5:\u00a0<\/strong>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/p>\n<p style=\"padding-left: 40px;\">The P-value [latex]=0.1928 > \\alpha = 0.05[\/latex]. Therefore, there is not enough evidence to reject the null hypothesis.<\/p>\n<p><strong>Step 6:\u00a0<\/strong>Write a conclusion in context.<\/p>\n<p style=\"padding-left: 40px;\">At the [latex]5\\%[\/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.<\/p>\n<p>Therefore, the community college&#8217;s beliefs are not correct.<\/p>\n<\/div>\n<\/div>\n<ul>\n<li>Using a confidence interval, does the data suggest that College A has taken more math classes than College B?<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q414734\">Show Answer<\/button><\/p>\n<div id=\"q414734\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the statistical tool, we find that the [latex]95\\%[\/latex] confidence interval for this data set is [latex](-0.683, 1.683)[\/latex].<\/p>\n<p>Interpretation: We are [latex]95\\%[\/latex] confident that the true difference in the mean math classes between College A and College B is [latex]-0.683[\/latex] and [latex]1.683[\/latex] classes.<\/p>\n<p>The values in the [latex]95\\%[\/latex] confidence interval include some negative values, some positive values, and [latex]0[\/latex]. This implies that we cannot conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.<\/p>\n<p>We achieved the same conclusion as the hypothesis test.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":25,"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":"fresh_take","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\/1335"}],"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\/1335\/revisions"}],"predecessor-version":[{"id":6821,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1335\/revisions\/6821"}],"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\/1335\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1335"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1335"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1335"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}