{"id":1275,"date":"2023-06-22T02:13:09","date_gmt":"2023-06-22T02:13:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-learn-it-3\/"},"modified":"2024-01-17T22:22:08","modified_gmt":"2024-01-17T22:22:08","slug":"sampling-distribution-of-a-sample-mean-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-mean-learn-it-3\/","title":{"raw":"Sampling Distribution of a Sample Mean: Apply It 2","rendered":"Sampling Distribution of a Sample Mean: Apply 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 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<p>When sampling from a normal population, such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean, but the variability in sample means will be less than the variability in individuals. This is similar to how there is less variability in sample proportions than the variability in individuals.<\/p>\r\n<section class=\"textbox recall\">\r\n<div>\r\n<p><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">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:<\/span><\/p>\r\n<\/div>\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 style=\"padding-left: 40px;\">where [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively.<\/p>\r\n<p>Note:<\/p>\r\n<div>\r\n<ul>\r\n\t<li>If the population distribution is normal, the distribution of the sample means will also follow a normal distribution, for any sample size.<\/li>\r\n\t<li>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.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1789[\/ohm2_question]<\/section>\r\n<p>Now that we have found the mean and standard deviation of the sampling distribution of the sample mean, we can use it to calculate probability and make inferences about the population.<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/ \" width=\"100%\" height=\"600\" 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\/normaldist\/\" 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]965[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]966[\/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<p>When sampling from a normal population, such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean, but the variability in sample means will be less than the variability in individuals. This is similar to how there is less variability in sample proportions than the variability in individuals.<\/p>\n<section class=\"textbox recall\">\n<div>\n<p><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">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:<\/span><\/p>\n<\/div>\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 style=\"padding-left: 40px;\">where [latex]\\mu[\/latex] and [latex]\\sigma[\/latex] represent the mean and standard deviation of the original population, respectively.<\/p>\n<p>Note:<\/p>\n<div>\n<ul>\n<li>If the population distribution is normal, the distribution of the sample means will also follow a normal distribution, for any sample size.<\/li>\n<li>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.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1789\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1789&theme=lumen&iframe_resize_id=ohm1789&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Now that we have found the mean and standard deviation of the sampling distribution of the sample mean, we can use it to calculate probability and make inferences about the population.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"600\" 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\/normaldist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm965\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=965&theme=lumen&iframe_resize_id=ohm965&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm966\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=966&theme=lumen&iframe_resize_id=ohm966&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":12,"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":"apply_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\/1275"}],"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\/1275\/revisions"}],"predecessor-version":[{"id":5121,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1275\/revisions\/5121"}],"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\/1275\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1275"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1275"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1275"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}