{"id":1281,"date":"2023-06-22T02:13:14","date_gmt":"2023-06-22T02:13:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-dig-deeper\/"},"modified":"2025-05-16T03:57:43","modified_gmt":"2025-05-16T03:57:43","slug":"sampling-distribution-of-a-sample-mean-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-dig-deeper\/","title":{"raw":"Sampling Distribution of a Sample Mean: Fresh Take","rendered":"Sampling Distribution of a Sample Mean: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Create a sampling distribution given [latex]\\mu[\/latex] and [latex]n[\/latex].<\/li>\r\n\t<li>Know and check the conditions of the Central Limit Theorem.<\/li>\r\n\t<li>Use the normal approximation to compute probabilities involving sample means when appropriate.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sampling Variability<\/h2>\r\n<p>Data collected by the Centers for Disease Control and Prevention show that the average birthweight for babies in the United States is [latex]7.17[\/latex] pounds, and the standard deviation of birthweights is [latex]1.30[\/latex] pounds[footnote]Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded). https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC[\/footnote]. Assume that birthweights in the United States follow an approximately normal distribution.<\/p>\r\n<p>Let's use the statistical tool to simulate random samples of births and examine the mean birthweight for each sample.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1:\u00a0<\/strong>Select Population Distribution: <strong>Bell-Shaped<\/strong>.<br \/>\r\n<strong>Step 2: <\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Enter [latex]7.17[\/latex] and [latex]1.30[\/latex] for the population mean and standard deviation, respectively. (You will need to select the <strong>Enter values for [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]<\/strong>\u00a0option.)<br \/>\r\n<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Step 3: <\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Select [latex]n=5[\/latex] and draw 1 sample.<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"><br \/>\r\nStep 4:\u00a0<\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Under the <strong>Data Distribution (Histogram from last generated sample)<\/strong>, you can find its [latex]\\bar(x)[\/latex] and [latex]s[\/latex].<\/span><\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_cont\/ \" width=\"100%\" height=\"1300\" 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\/sampdist_cont\/\" 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]1790[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1791[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1792[\/ohm2_question]<\/section>\r\n<p class=\"Para\">Because the birth weights of different babies will vary from sample to sample, the sample mean birth weights will also vary from sample to sample. The tendency of samples to have different statistics (means, proportions) than the population as a whole due to randomness is called <b>sampling variability<\/b>, and the distribution of these statistics is called a <b>sampling distribution<\/b>.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]968[\/ohm2_question]<\/section>\r\n<h2>Mean and Standard Deviation<\/h2>\r\n<p>In the case of sample means, if we sample from a normal population as the one seen here, the sampling distribution of the sample means will also have a normal distribution.<\/p>\r\n<section class=\"textbox recall\">If the mean and standard deviation of the population are [latex]\\mu[\/latex]\u00a0and [latex]\\sigma[\/latex], respectively, the mean and standard deviation of the sample means for random samples of size [latex]n[\/latex]\u00a0are:\r\n\r\n<ul>\r\n\t<li>Mean of the sample means = [latex]\\mu[\/latex]<\/li>\r\n\t<li>Standard deviation of the sample means = [latex]\\dfrac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1793[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Create a sampling distribution given [latex]\\mu[\/latex] and [latex]n[\/latex].<\/li>\n<li>Know and check the conditions of the Central Limit Theorem.<\/li>\n<li>Use the normal approximation to compute probabilities involving sample means when appropriate.<\/li>\n<\/ul>\n<\/section>\n<h2>Sampling Variability<\/h2>\n<p>Data collected by the Centers for Disease Control and Prevention show that the average birthweight for babies in the United States is [latex]7.17[\/latex] pounds, and the standard deviation of birthweights is [latex]1.30[\/latex] pounds<a class=\"footnote\" title=\"Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded). https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC\" id=\"return-footnote-1281-1\" href=\"#footnote-1281-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. Assume that birthweights in the United States follow an approximately normal distribution.<\/p>\n<p>Let&#8217;s use the statistical tool to simulate random samples of births and examine the mean birthweight for each sample.<\/p>\n<section class=\"textbox interact\"><strong>Step 1:\u00a0<\/strong>Select Population Distribution: <strong>Bell-Shaped<\/strong>.<br \/>\n<strong>Step 2: <\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Enter [latex]7.17[\/latex] and [latex]1.30[\/latex] for the population mean and standard deviation, respectively. (You will need to select the <strong>Enter values for [latex]\\mu[\/latex] and [latex]\\sigma[\/latex]<\/strong>\u00a0option.)<br \/>\n<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Step 3: <\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Select [latex]n=5[\/latex] and draw 1 sample.<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\"><br \/>\nStep 4:\u00a0<\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">Under the <strong>Data Distribution (Histogram from last generated sample)<\/strong>, you can find its [latex]\\bar(x)[\/latex] and [latex]s[\/latex].<\/span><\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/sampdist_cont\/\" width=\"100%\" height=\"1300\" 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\/sampdist_cont\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1790\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1790&theme=lumen&iframe_resize_id=ohm1790&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1791\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1791&theme=lumen&iframe_resize_id=ohm1791&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1792\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1792&theme=lumen&iframe_resize_id=ohm1792&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"Para\">Because the birth weights of different babies will vary from sample to sample, the sample mean birth weights will also vary from sample to sample. The tendency of samples to have different statistics (means, proportions) than the population as a whole due to randomness is called <b>sampling variability<\/b>, and the distribution of these statistics is called a <b>sampling distribution<\/b>.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm968\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=968&theme=lumen&iframe_resize_id=ohm968&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Mean and Standard Deviation<\/h2>\n<p>In the case of sample means, if we sample from a normal population as the one seen here, the sampling distribution of the sample means will also have a normal distribution.<\/p>\n<section class=\"textbox recall\">If the mean and standard deviation of the population are [latex]\\mu[\/latex]\u00a0and [latex]\\sigma[\/latex], respectively, the mean and standard deviation of the sample means for random samples of size [latex]n[\/latex]\u00a0are:<\/p>\n<ul>\n<li>Mean of the sample means = [latex]\\mu[\/latex]<\/li>\n<li>Standard deviation of the sample means = [latex]\\dfrac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1793\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1793&theme=lumen&iframe_resize_id=ohm1793&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1281-1\">Centers for Disease Control and Prevention. (n.d.). Natality for 2016\u20132019 (expanded). https:\/\/wonder.cdc.gov\/controller\/datarequest\/D149;jsessionid=7AB7525C7DC02FF1F19D38C125AC <a href=\"#return-footnote-1281-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":13,"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\/1281"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1281\/revisions"}],"predecessor-version":[{"id":6785,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1281\/revisions\/6785"}],"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\/1281\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1281"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1281"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1281"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1281"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}