{"id":1292,"date":"2023-06-22T02:13:23","date_gmt":"2023-06-22T02:13:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-a-population-mean-learn-it-1\/"},"modified":"2025-03-03T15:34:59","modified_gmt":"2025-03-03T15:34:59","slug":"confidence-interval-for-a-population-mean-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/confidence-interval-for-a-population-mean-learn-it-1\/","title":{"raw":"Confidence Interval for a Population Mean: Learn It 1","rendered":"Confidence Interval for a Population Mean: Learn It 1"},"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<p class=\"para\">When estimating a population mean, you usually start with data from a sample from the population of interest. To estimate the population mean, you start by calculating the sample mean. The sample mean can then be used to construct a confidence interval for the population mean in the same way that a sample proportion is used to construct a confidence interval estimate of a population proportion.<\/p>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p>The form of a confidence interval is <strong>estimate [latex]\\pm[\/latex] margin of error<\/strong>, where the margin of error was calculated by multiplying the standard error of the estimate by a [latex]z[\/latex]-critical value corresponding to the desired confidence level.<\/p>\r\n<\/section>\r\n<p>In a previous section, we constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions or conditions were met.\u00a0<\/p>\r\n<p>Let's look at the sampling distribution of the sample mean to help us construct a confidence interval to estimate a population mean when conditions are met and interpret the confidence interval in context.<\/p>\r\n<section class=\"textbox recall\">\r\n<div>\r\n<p><strong>Sampling Distribution of the Sample Mean<\/strong><\/p>\r\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and sample standard deviation [latex]s[\/latex]:<\/p>\r\n<ul>\r\n\t<li>The mean of the distribution of the sample means is [latex]\\mu[\/latex].<\/li>\r\n\t<li>The standard error of the distribution of the sample means is [latex]\\dfrac{s}{\\sqrt{n}}[\/latex].<\/li>\r\n\t<li>If the population distribution is normal or if the sample size is large ([latex]n \\ge 30[\/latex]), the distribution of the sample means follows an approximate normal distribution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1805[\/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<p class=\"para\">When estimating a population mean, you usually start with data from a sample from the population of interest. To estimate the population mean, you start by calculating the sample mean. The sample mean can then be used to construct a confidence interval for the population mean in the same way that a sample proportion is used to construct a confidence interval estimate of a population proportion.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p>The form of a confidence interval is <strong>estimate [latex]\\pm[\/latex] margin of error<\/strong>, where the margin of error was calculated by multiplying the standard error of the estimate by a [latex]z[\/latex]-critical value corresponding to the desired confidence level.<\/p>\n<\/section>\n<p>In a previous section, we constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions or conditions were met.\u00a0<\/p>\n<p>Let&#8217;s look at the sampling distribution of the sample mean to help us construct a confidence interval to estimate a population mean when conditions are met and interpret the confidence interval in context.<\/p>\n<section class=\"textbox recall\">\n<div>\n<p><strong>Sampling Distribution of the Sample Mean<\/strong><\/p>\n<p>When taking many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and sample standard deviation [latex]s[\/latex]:<\/p>\n<ul>\n<li>The mean of the distribution of the sample means is [latex]\\mu[\/latex].<\/li>\n<li>The standard error of the distribution of the sample means is [latex]\\dfrac{s}{\\sqrt{n}}[\/latex].<\/li>\n<li>If the population distribution is normal or if the sample size is large ([latex]n \\ge 30[\/latex]), the distribution of the sample means follows an approximate normal distribution.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1805\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1805&theme=lumen&iframe_resize_id=ohm1805&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":21,"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\/1292"}],"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\/1292\/revisions"}],"predecessor-version":[{"id":6325,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1292\/revisions\/6325"}],"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\/1292\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1292"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1292"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1292"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}