{"id":1511,"date":"2023-06-22T02:40:14","date_gmt":"2023-06-22T02:40:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-2\/"},"modified":"2025-05-17T02:50:11","modified_gmt":"2025-05-17T02:50:11","slug":"bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/bootstrap-distribution-and-confidence-interval-for-a-population-mean-learn-it-2\/","title":{"raw":"Bootstrap Distribution and Confidence Interval for a Population Mean - Learn It 2","rendered":"Bootstrap Distribution and Confidence Interval for a Population Mean &#8211; 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;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<h2>Bootstrap Sample<\/h2>\r\n<p>Let's learn a new way to construct confidence intervals that involves resampling from the data in a sample from a population. This involves selecting bootstrap samples.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>bootstrap sample<\/strong><\/h3>\r\n<p>A <strong>bootstrap sample<\/strong> is a sample that is selected from the values in the original sample.<\/p>\r\n<p>The bootstrap sample is selected with replacement, and the sample size is the same as the sample size of the original sample.<\/p>\r\n<\/section>\r\n<p>Let's practice selecting a bootstrap sample from the sample of the [latex]10[\/latex] electric car kWh\/[latex]100[\/latex] miles ratings.<\/p>\r\n<p>To create a bootstrap sample from the original sample, we want to select a new sample of size [latex]10[\/latex] by sampling with replacement from the original sample. Notice that the observations in the original sample are numbered from [latex]1[\/latex] to [latex]10[\/latex]. To select the bootstrap sample, we can generate [latex]10[\/latex] random numbers between [latex]1[\/latex] and [latex]10[\/latex] to identify which values are included in the bootstrap sample. Because we are sampling with replacement, we should expect that some of the random numbers generated may be repeated.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Enter \u201c1\u201d in the \u201cChoose Minimum\u201d box and \u201c10\u201d in the \u201cChoose Maximum\u201d box.<br \/>\r\n<strong>Step 2: <\/strong>Enter \u201c10\u201d in the \u201cHow many numbers do you want to generate?\u201d box.<br \/>\r\n<strong>Step 3: <\/strong>Select \u201cYes\u201d for \u201cSample with Replacement?\u201d<br \/>\r\n<strong>Step 4: <\/strong>Click on the \u201cGenerate\u201d button, and the app will display the 10 numbers generated in the box labeled \u201cCurrent Simulation.\u201d<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" width=\"100%\" height=\"850\" frameborder=\"no\"><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3112[\/ohm2_question]<\/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<h2>Bootstrap Sample<\/h2>\n<p>Let&#8217;s learn a new way to construct confidence intervals that involves resampling from the data in a sample from a population. This involves selecting bootstrap samples.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>bootstrap sample<\/strong><\/h3>\n<p>A <strong>bootstrap sample<\/strong> is a sample that is selected from the values in the original sample.<\/p>\n<p>The bootstrap sample is selected with replacement, and the sample size is the same as the sample size of the original sample.<\/p>\n<\/section>\n<p>Let&#8217;s practice selecting a bootstrap sample from the sample of the [latex]10[\/latex] electric car kWh\/[latex]100[\/latex] miles ratings.<\/p>\n<p>To create a bootstrap sample from the original sample, we want to select a new sample of size [latex]10[\/latex] by sampling with replacement from the original sample. Notice that the observations in the original sample are numbered from [latex]1[\/latex] to [latex]10[\/latex]. To select the bootstrap sample, we can generate [latex]10[\/latex] random numbers between [latex]1[\/latex] and [latex]10[\/latex] to identify which values are included in the bootstrap sample. Because we are sampling with replacement, we should expect that some of the random numbers generated may be repeated.<\/p>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Enter \u201c1\u201d in the \u201cChoose Minimum\u201d box and \u201c10\u201d in the \u201cChoose Maximum\u201d box.<br \/>\n<strong>Step 2: <\/strong>Enter \u201c10\u201d in the \u201cHow many numbers do you want to generate?\u201d box.<br \/>\n<strong>Step 3: <\/strong>Select \u201cYes\u201d for \u201cSample with Replacement?\u201d<br \/>\n<strong>Step 4: <\/strong>Click on the \u201cGenerate\u201d button, and the app will display the 10 numbers generated in the box labeled \u201cCurrent Simulation.\u201d<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" width=\"100%\" height=\"850\" frameborder=\"no\"><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/randomnumbers\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3112\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3112&theme=lumen&iframe_resize_id=ohm3112&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":9,"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\/1511"}],"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\/1511\/revisions"}],"predecessor-version":[{"id":6923,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1511\/revisions\/6923"}],"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\/1511\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1511"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1511"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1511"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}