{"id":1507,"date":"2023-06-22T02:40:12","date_gmt":"2023-06-22T02:40:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-18-background-youll-need-4\/"},"modified":"2025-05-17T02:48:56","modified_gmt":"2025-05-17T02:48:56","slug":"module-18-background-youll-need-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-18-background-youll-need-4\/","title":{"raw":"Module 17: Background You'll Need 4","rendered":"Module 17: Background You&#8217;ll Need 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;Know how confidence intervals for a difference in means are interpreted&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12928,&quot;10&quot;:2,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Know how confidence intervals for a difference in means are interpreted<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2><strong>Interpreting Confidence Intervals for a Difference in Means<\/strong><\/h2>\r\n<p>Recall that a confidence interval for a difference in population means is interpreted as an interval of plausible values for the difference in means.<\/p>\r\n<section class=\"textbox example\">Suppose that each student in a random sample of [latex]50[\/latex] first-year students at a college and each student in a random sample of [latex]50[\/latex] second-year students from the college are asked how many hours of sleep they get on a typical weekday night. The data from these two samples are used to construct a confidence interval for the difference [latex]\\mu_F - \\mu_S[\/latex], where [latex]\\mu_F[\/latex] is the mean number of sleep hours for first-year students and [latex]\\mu_S[\/latex] is the mean number of sleep hours for second-year students. Because the samples are random samples and the sample sizes are both greater than [latex]30[\/latex], it would be appropriate to use a two-sample [latex]t[\/latex] confidence interval.\r\n\r\n<p>If the [latex]95\\%[\/latex] confidence interval was [latex](0.4, 1.0)[\/latex], we would note the following:<\/p>\r\n<ul>\r\n\t<li>Plausible values for are between [latex]0.4[\/latex] and [latex]1.0[\/latex].<\/li>\r\n\t<li>All of the plausible values are positive, which corresponds to [latex]\\mu_F[\/latex] being greater than [latex]\\mu_S[\/latex].<\/li>\r\n\t<li>We are [latex]95\\%[\/latex] confident that the mean number of hours of sleep for first-year students is greater than the mean number of hours of sleep for second-year students by somewhere between [latex]0.4[\/latex] and [latex]1.0[\/latex] hours.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">[ohm2_question hide_question_numbers=1]3494[\/ohm2_question]<\/span><\/section>\r\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">[ohm2_question hide_question_numbers=1]3495[\/ohm2_question]<\/span><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Know how confidence intervals for a difference in means are interpreted&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12928,&quot;10&quot;:2,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Know how confidence intervals for a difference in means are interpreted<\/span><\/li>\n<\/ul>\n<\/section>\n<h2><strong>Interpreting Confidence Intervals for a Difference in Means<\/strong><\/h2>\n<p>Recall that a confidence interval for a difference in population means is interpreted as an interval of plausible values for the difference in means.<\/p>\n<section class=\"textbox example\">Suppose that each student in a random sample of [latex]50[\/latex] first-year students at a college and each student in a random sample of [latex]50[\/latex] second-year students from the college are asked how many hours of sleep they get on a typical weekday night. The data from these two samples are used to construct a confidence interval for the difference [latex]\\mu_F - \\mu_S[\/latex], where [latex]\\mu_F[\/latex] is the mean number of sleep hours for first-year students and [latex]\\mu_S[\/latex] is the mean number of sleep hours for second-year students. Because the samples are random samples and the sample sizes are both greater than [latex]30[\/latex], it would be appropriate to use a two-sample [latex]t[\/latex] confidence interval.<\/p>\n<p>If the [latex]95\\%[\/latex] confidence interval was [latex](0.4, 1.0)[\/latex], we would note the following:<\/p>\n<ul>\n<li>Plausible values for are between [latex]0.4[\/latex] and [latex]1.0[\/latex].<\/li>\n<li>All of the plausible values are positive, which corresponds to [latex]\\mu_F[\/latex] being greater than [latex]\\mu_S[\/latex].<\/li>\n<li>We are [latex]95\\%[\/latex] confident that the mean number of hours of sleep for first-year students is greater than the mean number of hours of sleep for second-year students by somewhere between [latex]0.4[\/latex] and [latex]1.0[\/latex] hours.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"><iframe loading=\"lazy\" id=\"ohm3494\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3494&theme=lumen&iframe_resize_id=ohm3494&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n<section class=\"textbox tryIt\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"><iframe loading=\"lazy\" id=\"ohm3495\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3495&theme=lumen&iframe_resize_id=ohm3495&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n","protected":false},"author":8,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1502,"module-header":"background_you_need","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\/1507"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1507\/revisions"}],"predecessor-version":[{"id":6919,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1507\/revisions\/6919"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1502"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1507\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1507"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1507"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1507"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}