{"id":1305,"date":"2023-06-22T02:13:33","date_gmt":"2023-06-22T02:13:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-apply-it-2\/"},"modified":"2025-05-16T04:06:42","modified_gmt":"2025-05-16T04:06:42","slug":"confidence-interval-for-difference-in-population-means-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-difference-in-population-means-fresh-take\/","title":{"raw":"Confidence Interval for Difference in Population Means: Fresh Take","rendered":"Confidence Interval for Difference in Population Means: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/li>\r\n\t<li class=\"li1\">Calculate and explain a confidence interval for the difference between two population means.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>If it's hard to read, it's hard to do!<\/h2>\r\n<p><img class=\"aligncenter wp-image-741 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/23211700\/Picture1-8.png\" alt=\"This is not hard to read written in Arial font. This is hard to read written in Mistral font.\" width=\"300\" height=\"90\" \/><\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]978[\/ohm2_question]<\/section>\r\n<section class=\"textbox recall\">\r\n<p class=\"studentredanswer\" style=\"margin-left: 0in;\"><span style=\"color: windowtext;\">There are a few assumptions\/conditions that you should check before using the two-sample [latex]t[\/latex] confidence interval. These were introduced in the preview assignment and are included here as a reminder:<\/span><\/p>\r\n<ol>\r\n\t<li class=\"studentredanswer\"><span style=\"color: windowtext;\">The samples are independent.<\/span><\/li>\r\n\t<li class=\"studentredanswer\"><span style=\"color: windowtext;\">Each sample is a random sample from the corresponding population of interest, or it is reasonable to regard the sample as if it were a random sample. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in the sample being representative of the population. <b>If the data are from an experiment, you just need to check that there was random assignment to experimental groups\u2014this substitutes for the random sample condition and also results in independent samples.<\/b><\/span><\/li>\r\n\t<li class=\"studentredanswer\"><span style=\"color: windowtext;\">For each population, the distribution of the variable that was measured is approximately normal, or the sample size for the sample from that population is large. 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.<\/span><\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox recall\">The confidence interval for difference in means is:[latex](\\bar{x}_1-\\bar{x}_2) \\pm (t - \\text{critical value})\\cdot \\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}[\/latex]<\/section>\r\n<p>As we learned previously, the [latex]t[\/latex]-distribution depends on the <strong>degrees of freedom ([latex]df[\/latex])<\/strong>. In the one-sample cases, [latex]df\u00a0=\u00a0n \u2013 1[\/latex]. For the two-sample [latex]t[\/latex] confidence interval, determining the correct [latex]df[\/latex] is based on a complicated formula that we do not cover in this course. We will either give the [latex]df[\/latex]\u00a0or use technology to find the\u00a0[latex]df[\/latex] and the [latex]t[\/latex]-critical value.<\/p>\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<p>[reveal-answer q=\"801848\"]Steps to use the statistical tool.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"801848\"]<\/p>\r\n<ul>\r\n\t<li>Select <strong>Summary Statistics<\/strong> from the drop-down menu under <strong>Enter Data<\/strong>.<\/li>\r\n\t<li>Type in <strong>Estimated Time to Complete<\/strong> for the name of the variable, and type in <strong>Easy to Read<\/strong> for the Group 1 label and <strong>Hard to Read<\/strong>\u00a0for the Group 2 label.<\/li>\r\n\t<li>Enter the sample sizes and the sample means and standard deviations for this example.<\/li>\r\n\t<li>For <strong>Type of Inference<\/strong>, select <strong>Confidence Interval<\/strong>.<\/li>\r\n\t<li>Use the slider for the confidence level to select a 90% confidence level.<\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1808[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Check the assumptions for a two-sample [latex]t[\/latex] confidence interval for population mean.<\/li>\n<li class=\"li1\">Calculate and explain a confidence interval for the difference between two population means.<\/li>\n<\/ul>\n<\/section>\n<h2>If it&#8217;s hard to read, it&#8217;s hard to do!<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-741 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/23211700\/Picture1-8.png\" alt=\"This is not hard to read written in Arial font. This is hard to read written in Mistral font.\" width=\"300\" height=\"90\" \/><\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm978\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=978&theme=lumen&iframe_resize_id=ohm978&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\">\n<p class=\"studentredanswer\" style=\"margin-left: 0in;\"><span style=\"color: windowtext;\">There are a few assumptions\/conditions that you should check before using the two-sample [latex]t[\/latex] confidence interval. These were introduced in the preview assignment and are included here as a reminder:<\/span><\/p>\n<ol>\n<li class=\"studentredanswer\"><span style=\"color: windowtext;\">The samples are independent.<\/span><\/li>\n<li class=\"studentredanswer\"><span style=\"color: windowtext;\">Each sample is a random sample from the corresponding population of interest, or it is reasonable to regard the sample as if it were a random sample. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in the sample being representative of the population. <b>If the data are from an experiment, you just need to check that there was random assignment to experimental groups\u2014this substitutes for the random sample condition and also results in independent samples.<\/b><\/span><\/li>\n<li class=\"studentredanswer\"><span style=\"color: windowtext;\">For each population, the distribution of the variable that was measured is approximately normal, or the sample size for the sample from that population is large. 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.<\/span><\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox recall\">The confidence interval for difference in means is:[latex](\\bar{x}_1-\\bar{x}_2) \\pm (t - \\text{critical value})\\cdot \\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}[\/latex]<\/section>\n<p>As we learned previously, the [latex]t[\/latex]-distribution depends on the <strong>degrees of freedom ([latex]df[\/latex])<\/strong>. In the one-sample cases, [latex]df\u00a0=\u00a0n \u2013 1[\/latex]. For the two-sample [latex]t[\/latex] confidence interval, determining the correct [latex]df[\/latex] is based on a complicated formula that we do not cover in this course. We will either give the [latex]df[\/latex]\u00a0or use technology to find the\u00a0[latex]df[\/latex] and the [latex]t[\/latex]-critical value.<\/p>\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q801848\">Steps to use the statistical tool.<\/button><\/p>\n<div id=\"q801848\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Select <strong>Summary Statistics<\/strong> from the drop-down menu under <strong>Enter Data<\/strong>.<\/li>\n<li>Type in <strong>Estimated Time to Complete<\/strong> for the name of the variable, and type in <strong>Easy to Read<\/strong> for the Group 1 label and <strong>Hard to Read<\/strong>\u00a0for the Group 2 label.<\/li>\n<li>Enter the sample sizes and the sample means and standard deviations for this example.<\/li>\n<li>For <strong>Type of Inference<\/strong>, select <strong>Confidence Interval<\/strong>.<\/li>\n<li>Use the slider for the confidence level to select a 90% confidence level.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1808\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1808&theme=lumen&iframe_resize_id=ohm1808&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":34,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1268,"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\/1305"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1305\/revisions"}],"predecessor-version":[{"id":6800,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1305\/revisions\/6800"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1268"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1305\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1305"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1305"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1305"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}