{"id":1279,"date":"2023-06-22T02:13:12","date_gmt":"2023-06-22T02:13:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-apply-it-3\/"},"modified":"2024-01-17T22:00:08","modified_gmt":"2024-01-17T22:00:08","slug":"sampling-distribution-of-a-sample-mean-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-apply-it-3\/","title":{"raw":"Sampling Distribution of a Sample Mean: Learn It 3","rendered":"Sampling Distribution of a Sample Mean: Learn It 3"},"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 sampling distribution given [latex]\\\\mu[\/latex] and [latex]n[\/latex]&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;}\">Create a sampling distribution given [latex]\\mu[\/latex] and [latex]n[\/latex].<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Know and check the conditions of the Central Limit Theorem&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;}\">Know and check the conditions of the Central Limit Theorem.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal approximation to compute probabilities involving sample means when appropriate&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;}\">Use the normal approximation to compute probabilities involving sample means when appropriate.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>mean and standard deviation of the sampling distribution<\/h3>\r\n<p>The mathematical formulas to find the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[\/latex] are:<\/p>\r\n<ul>\r\n\t<li><strong>Mean<\/strong> of the sampling distribution of the sample mean [latex]=\\mu[\/latex]<\/li>\r\n\t<li><strong>Standard deviation<\/strong> of the sampling distribution of the sample mean [latex]=\\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<p>where [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1501[\/ohm2_question]<\/section>\r\n<p>You should notice that the mean and standard deviation calculated above is nearly the same as the mean and standard deviation found using the simulation of [latex]1,000[\/latex] random samples. You just witnessed the Central Limit Theorem at work for sample means. The <strong>Central Limit Theorem<\/strong> states that, as the sample size gets larger, the distribution of the sample mean will become closer to a normal distribution.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Central Limit Theorem<\/h3>\r\n<p>If the population distribution is normal, the distribution of the sample means will also follow a normal distribution, for any sample size.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>If the population distribution is not normal, the <strong>Central Limit Theorem<\/strong> states that the distribution of the sample means still follows an approximate normal distribution as long as the sample size is large (e.g., [latex]n \\ge 30[\/latex]) and the population distribution is not strongly skewed.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1897[\/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 sampling distribution given [latex]\\\\mu[\/latex] and [latex]n[\/latex]&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;}\">Create a sampling distribution given [latex]\\mu[\/latex] and [latex]n[\/latex].<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Know and check the conditions of the Central Limit Theorem&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;}\">Know and check the conditions of the Central Limit Theorem.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the normal approximation to compute probabilities involving sample means when appropriate&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;}\">Use the normal approximation to compute probabilities involving sample means when appropriate.<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>mean and standard deviation of the sampling distribution<\/h3>\n<p>The mathematical formulas to find the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[\/latex] are:<\/p>\n<ul>\n<li><strong>Mean<\/strong> of the sampling distribution of the sample mean [latex]=\\mu[\/latex]<\/li>\n<li><strong>Standard deviation<\/strong> of the sampling distribution of the sample mean [latex]=\\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\n<\/ul>\n<p>where [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1501\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1501&theme=lumen&iframe_resize_id=ohm1501&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>You should notice that the mean and standard deviation calculated above is nearly the same as the mean and standard deviation found using the simulation of [latex]1,000[\/latex] random samples. You just witnessed the Central Limit Theorem at work for sample means. The <strong>Central Limit Theorem<\/strong> states that, as the sample size gets larger, the distribution of the sample mean will become closer to a normal distribution.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Central Limit Theorem<\/h3>\n<p>If the population distribution is normal, the distribution of the sample means will also follow a normal distribution, for any sample size.<\/p>\n<p>&nbsp;<\/p>\n<p>If the population distribution is not normal, the <strong>Central Limit Theorem<\/strong> states that the distribution of the sample means still follows an approximate normal distribution as long as the sample size is large (e.g., [latex]n \\ge 30[\/latex]) and the population distribution is not strongly skewed.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1897\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1897&theme=lumen&iframe_resize_id=ohm1897&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":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\/1279"}],"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\/1279\/revisions"}],"predecessor-version":[{"id":5117,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1279\/revisions\/5117"}],"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\/1279\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1279"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1279"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1279"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}