{"id":1293,"date":"2023-06-22T02:13:24","date_gmt":"2023-06-22T02:13:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-a-population-mean-learn-it-2\/"},"modified":"2025-05-16T04:01:22","modified_gmt":"2025-05-16T04:01:22","slug":"confidence-interval-for-a-population-mean-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-a-population-mean-learn-it-2\/","title":{"raw":"Confidence Interval for a Population Mean: Learn It 2","rendered":"Confidence Interval for a Population Mean: 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;Check the assumptions for a one-sample t confidence interval for population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the assumptions for a one-sample [latex]t[\/latex] confidence interval for population mean.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval for a population mean and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate a confidence interval for a population mean and explain what it means.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Confidence Interval for a Population Mean<\/h2>\r\n<section class=\"textbox recall\">Confidence interval can be calculated using: <strong>estimate [latex]\\pm[\/latex] margin of error.<\/strong><\/section>\r\n<p>The margin of error is calculated a little differently: Instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate [latex]t[\/latex]-distribution.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>confidence interval for a population mean<\/h3>\r\n<p>The formula for a <strong>confidence interval for a population mean<\/strong> is:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\bar{x} \\pm (t\\text{-critical value})\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The [latex]t[\/latex]-critical value in the confidence interval will depend on the sample size (degrees of freedom for the [latex]t[\/latex]-distribution: [latex]df=n-1[\/latex]) and the confidence level.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>This interval is often called a <strong>one-sample [latex]t[\/latex] interval<\/strong>.<\/p>\r\n<\/section>\r\n<section>Let's utilize technology to help us find the confidence interval for a population mean.\u00a0<\/section>\r\n<section class=\"textbox interact\"><b>Step 1:<\/b> Under <strong>Enter Data<\/strong>, select <strong>Summary Statistics<\/strong>.<br \/>\r\n<strong>Step 2:\u00a0<\/strong>Enter the <strong>Name of Variable<\/strong>, <strong>Sample Size<\/strong>, <strong>Sample Mean<\/strong>, and <strong>Sample Std. Dev.<\/strong>\u00a0accordingly.<br \/>\r\n<strong>Step 3:\u00a0<\/strong>Adjust the <strong>Confidence Level<\/strong>\u00a0based on your question.<\/section>\r\n<section><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/ \" width=\"100%\" height=\"1075\" 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\/inference_mean\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1916[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the assumptions for a one-sample t confidence interval for population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Check the assumptions for a one-sample [latex]t[\/latex] confidence interval for population mean.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval for a population mean and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4865,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate a confidence interval for a population mean and explain what it means.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Confidence Interval for a Population Mean<\/h2>\n<section class=\"textbox recall\">Confidence interval can be calculated using: <strong>estimate [latex]\\pm[\/latex] margin of error.<\/strong><\/section>\n<p>The margin of error is calculated a little differently: Instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate [latex]t[\/latex]-distribution.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>confidence interval for a population mean<\/h3>\n<p>The formula for a <strong>confidence interval for a population mean<\/strong> is:<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x} \\pm (t\\text{-critical value})\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The [latex]t[\/latex]-critical value in the confidence interval will depend on the sample size (degrees of freedom for the [latex]t[\/latex]-distribution: [latex]df=n-1[\/latex]) and the confidence level.<\/p>\n<p>&nbsp;<\/p>\n<p>This interval is often called a <strong>one-sample [latex]t[\/latex] interval<\/strong>.<\/p>\n<\/section>\n<section>Let&#8217;s utilize technology to help us find the confidence interval for a population mean.\u00a0<\/section>\n<section class=\"textbox interact\"><b>Step 1:<\/b> Under <strong>Enter Data<\/strong>, select <strong>Summary Statistics<\/strong>.<br \/>\n<strong>Step 2:\u00a0<\/strong>Enter the <strong>Name of Variable<\/strong>, <strong>Sample Size<\/strong>, <strong>Sample Mean<\/strong>, and <strong>Sample Std. Dev.<\/strong>\u00a0accordingly.<br \/>\n<strong>Step 3:\u00a0<\/strong>Adjust the <strong>Confidence Level<\/strong>\u00a0based on your question.<\/section>\n<section><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/\" width=\"100%\" height=\"1075\" 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\/inference_mean\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1916\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1916&theme=lumen&iframe_resize_id=ohm1916&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":22,"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":"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\/1293"}],"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\/1293\/revisions"}],"predecessor-version":[{"id":6792,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1293\/revisions\/6792"}],"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\/1293\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1293"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1293"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1293"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}