{"id":1514,"date":"2023-06-22T02:40:15","date_gmt":"2023-06-22T02:40:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-4\/"},"modified":"2024-02-08T23:33:29","modified_gmt":"2024-02-08T23:33:29","slug":"bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-4\/","title":{"raw":"Bootstrap Distribution and Confidence Interval for a Population Mean - Learn It 4","rendered":"Bootstrap Distribution and Confidence Interval for a Population Mean &#8211; Learn It 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;Create a bootstrap distribution for a sample mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Create a bootstrap distribution for a sample mean<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find and describe a bootstrap percentile confidence interval for a population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find and describe a bootstrap percentile confidence interval for a population mean<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"student12ptnumberlist\">To create a bootstrap distribution, we want to take a large number of bootstrap samples. So, we turn to technology. A statistics software package was used to create the following histogram of the bootstrap distribution based on [latex]1,000[\/latex] bootstrap samples:<\/p>\r\n<p><img class=\"aligncenter wp-image-2380 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22024014\/Picture1-2.png\" alt=\"Example bootstrap distribution of 1000 samples, with a mean of 34.\" width=\"685\" height=\"460\" \/><\/p>\r\n<p>The bootstrap distribution is used to obtain a confidence interval for the population mean. Notice that the bootstrap distribution is centered at [latex]34[\/latex], which was the sample mean of the original sample.<\/p>\r\n<section class=\"textbox proTip\">For a [latex]95\\%[\/latex] confidence level, using the boundaries that capture the middle [latex]95\\%[\/latex] of the bootstrap distribution to determine the endpoints of the confidence interval is equivalent to adding a number to the original sample mean and subtracting a number from the original sample mean. For a bootstrap confidence interval, the number that is added and the number that is subtracted are not the same because the bootstrap distribution is not symmetric.<\/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]3487[\/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;Create a bootstrap distribution for a sample mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Create a bootstrap distribution for a sample mean<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find and describe a bootstrap percentile confidence interval for a population mean&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find and describe a bootstrap percentile confidence interval for a population mean<\/span><\/li>\n<\/ul>\n<\/section>\n<p class=\"student12ptnumberlist\">To create a bootstrap distribution, we want to take a large number of bootstrap samples. So, we turn to technology. A statistics software package was used to create the following histogram of the bootstrap distribution based on [latex]1,000[\/latex] bootstrap samples:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2380 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22024014\/Picture1-2.png\" alt=\"Example bootstrap distribution of 1000 samples, with a mean of 34.\" width=\"685\" height=\"460\" \/><\/p>\n<p>The bootstrap distribution is used to obtain a confidence interval for the population mean. Notice that the bootstrap distribution is centered at [latex]34[\/latex], which was the sample mean of the original sample.<\/p>\n<section class=\"textbox proTip\">For a [latex]95\\%[\/latex] confidence level, using the boundaries that capture the middle [latex]95\\%[\/latex] of the bootstrap distribution to determine the endpoints of the confidence interval is equivalent to adding a number to the original sample mean and subtracting a number from the original sample mean. For a bootstrap confidence interval, the number that is added and the number that is subtracted are not the same because the bootstrap distribution is not symmetric.<\/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=\"ohm3487\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3487&theme=lumen&iframe_resize_id=ohm3487&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/span><\/section>\n","protected":false},"author":8,"menu_order":11,"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":"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\/1514"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1514\/revisions"}],"predecessor-version":[{"id":5506,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1514\/revisions\/5506"}],"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\/1514\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1514"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1514"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1514"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}