{"id":1298,"date":"2023-06-22T02:13:28","date_gmt":"2023-06-22T02:13:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-population-mean-dig-deeper\/"},"modified":"2024-01-19T17:58:59","modified_gmt":"2024-01-19T17:58:59","slug":"confidence-interval-for-population-mean-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-population-mean-dig-deeper\/","title":{"raw":"Confidence Interval for Population Mean: Fresh Take","rendered":"Confidence Interval for Population Mean: 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;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<section class=\"textbox recall\">The formula for a confidence interval for a population mean is:\r\n\r\n<p style=\"text-align: center;\">[latex]\\text{estimate }\\pm \\text{ margin of error}[\/latex]<\/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>where [latex]\\bar{x}[\/latex] is the sample mean and the standard error used is the standard error of the sample mean, [latex]\\frac{s}{\\sqrt{n}}[\/latex].<\/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<\/section>\r\n<section class=\"textbox example\">Using the statistical tool below, find the [latex]t\\text{-critical value}[\/latex] for a:<br \/>\r\na) [latex]95\\%[\/latex] confidence interval with a random sample of 45.<br \/>\r\nb) [latex]90\\%[\/latex] confidence interval with a random sample of 45.[reveal-answer q=\"484716\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"484716\"]<br \/>\r\nIf [latex]n=45[\/latex], then [latex]df = 45-1=44[\/latex].<br \/>\r\nUsing the \"Find Percentile\/Quartile\" tab with a [latex]df=44[\/latex] and a \"Two-Tailed\" type of percentile, we found that:\r\n\r\n<p>a) [latex]t\\text{-critical value}[\/latex] for a [latex]95\\%[\/latex] is [latex]\\pm 2.015[\/latex]<br \/>\r\nb) [latex]t\\text{-critical value}[\/latex] for a [latex]90\\%[\/latex] is [latex]\\pm 1.68[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/tdist\/ \" width=\"100%\" height=\"700\" 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\/tdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox example\">A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging. The engineers take a random sample of [latex]45[\/latex] cables and apply weights to each of them until they break. The mean breaking weight for the [latex]45[\/latex] cables is [latex]768.2[\/latex] lb. The standard deviation of the breaking weight for the sample is [latex]15.1[\/latex] lb. What should the engineers report as the mean amount of weight held by this type of cable?\r\n\r\n<p>Let\u2019s use these sample statistics to construct a [latex]95\\%[\/latex] confidence interval for the mean breaking weight of this type of cable.<\/p>\r\n<p>[reveal-answer q=\"143303\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"143303\"]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{estimate }\\pm \\text{ margin of error}[\/latex]<\/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 style=\"text-align: center;\">[latex]768.2 \\pm (2.015)\\frac{15.1}{\\sqrt{45}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]768.2 \\pm 4.53 = (763.7, 772.7)[\/latex]<\/p>\r\n<p><strong>Conclusion:<\/strong><\/p>\r\n<p>We are [latex]95\\%[\/latex] confident that the mean breaking weight for all cables of this type is between [latex]763.7[\/latex] lb and [latex]772.7[\/latex] lb.<\/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;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<section class=\"textbox recall\">The formula for a confidence interval for a population mean is:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{estimate }\\pm \\text{ margin of error}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x} \\pm (t\\text{-critical value})\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\n<p>where [latex]\\bar{x}[\/latex] is the sample mean and the standard error used is the standard error of the sample mean, [latex]\\frac{s}{\\sqrt{n}}[\/latex].<\/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<\/section>\n<section class=\"textbox example\">Using the statistical tool below, find the [latex]t\\text{-critical value}[\/latex] for a:<br \/>\na) [latex]95\\%[\/latex] confidence interval with a random sample of 45.<br \/>\nb) [latex]90\\%[\/latex] confidence interval with a random sample of 45.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q484716\">Show Answer<\/button><\/p>\n<div id=\"q484716\" class=\"hidden-answer\" style=\"display: none\">\nIf [latex]n=45[\/latex], then [latex]df = 45-1=44[\/latex].<br \/>\nUsing the &#8220;Find Percentile\/Quartile&#8221; tab with a [latex]df=44[\/latex] and a &#8220;Two-Tailed&#8221; type of percentile, we found that:<\/p>\n<p>a) [latex]t\\text{-critical value}[\/latex] for a [latex]95\\%[\/latex] is [latex]\\pm 2.015[\/latex]<br \/>\nb) [latex]t\\text{-critical value}[\/latex] for a [latex]90\\%[\/latex] is [latex]\\pm 1.68[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/tdist\/\" width=\"100%\" height=\"700\" 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\/tdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox example\">A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging. The engineers take a random sample of [latex]45[\/latex] cables and apply weights to each of them until they break. The mean breaking weight for the [latex]45[\/latex] cables is [latex]768.2[\/latex] lb. The standard deviation of the breaking weight for the sample is [latex]15.1[\/latex] lb. What should the engineers report as the mean amount of weight held by this type of cable?<\/p>\n<p>Let\u2019s use these sample statistics to construct a [latex]95\\%[\/latex] confidence interval for the mean breaking weight of this type of cable.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q143303\">Show answer<\/button><\/p>\n<div id=\"q143303\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\text{estimate }\\pm \\text{ margin of error}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\bar{x} \\pm (t\\text{-critical value})\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]768.2 \\pm (2.015)\\frac{15.1}{\\sqrt{45}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]768.2 \\pm 4.53 = (763.7, 772.7)[\/latex]<\/p>\n<p><strong>Conclusion:<\/strong><\/p>\n<p>We are [latex]95\\%[\/latex] confident that the mean breaking weight for all cables of this type is between [latex]763.7[\/latex] lb and [latex]772.7[\/latex] lb.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":27,"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\/1298"}],"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":3,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1298\/revisions"}],"predecessor-version":[{"id":5162,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1298\/revisions\/5162"}],"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\/1298\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1298"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1298"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1298"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}