{"id":1284,"date":"2023-06-22T02:13:16","date_gmt":"2023-06-22T02:13:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/t-distribution-learn-it-2\/"},"modified":"2025-05-16T03:58:18","modified_gmt":"2025-05-16T03:58:18","slug":"t-distribution-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/t-distribution-learn-it-2\/","title":{"raw":"t-distribution: Learn It 2","rendered":"t-distribution: 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;Check the conditions for a t-distribution, then use a t-distribution to calculate probabilities 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;}\">Check the conditions for a [latex]t[\/latex]-distribution, then use a [latex]t[\/latex]-distribution to calculate probabilities when appropriate.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Standardized Statistic<\/h2>\r\n<p>When we calculate a [latex]z[\/latex]-score for a statistic using simulation to estimate the mean and standard deviation of the sample mean, we call this a\u00a0<strong>standardized statistic<\/strong>. Mathematically, though, we know the exact formulas for these values.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>standardized statistic ([latex]z[\/latex])<\/strong><\/h3>\r\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. deviation of } \\bar{x}'s} [\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}[\/latex]<\/p>\r\n<p>where [latex]\\bar{x}[\/latex] is the sample mean, [latex]\\mu[\/latex] is the population mean, [latex]\\sigma[\/latex] is the population standard deviation, and [latex]n[\/latex] is the sample size. The statistic is \u201cstandardized\u201d since it is centered to have a mean of [latex]0[\/latex] and scaled to have a standard deviation of [latex]1[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p class=\"para\">If the population distribution is normal or the sample size is sufficiently large, this standardized statistic will follow a <b>standard normal distribution<\/b>: a normal distribution with a mean of [latex]0[\/latex] and a standard deviation of [latex]1[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1799[\/ohm2_question]<\/section>\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]975[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Check the conditions for a t-distribution, then use a t-distribution to calculate probabilities 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;}\">Check the conditions for a [latex]t[\/latex]-distribution, then use a [latex]t[\/latex]-distribution to calculate probabilities when appropriate.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Standardized Statistic<\/h2>\n<p>When we calculate a [latex]z[\/latex]-score for a statistic using simulation to estimate the mean and standard deviation of the sample mean, we call this a\u00a0<strong>standardized statistic<\/strong>. Mathematically, though, we know the exact formulas for these values.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>standardized statistic ([latex]z[\/latex])<\/strong><\/h3>\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. deviation of } \\bar{x}'s}[\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}[\/latex]<\/p>\n<p>where [latex]\\bar{x}[\/latex] is the sample mean, [latex]\\mu[\/latex] is the population mean, [latex]\\sigma[\/latex] is the population standard deviation, and [latex]n[\/latex] is the sample size. The statistic is \u201cstandardized\u201d since it is centered to have a mean of [latex]0[\/latex] and scaled to have a standard deviation of [latex]1[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p class=\"para\">If the population distribution is normal or the sample size is sufficiently large, this standardized statistic will follow a <b>standard normal distribution<\/b>: a normal distribution with a mean of [latex]0[\/latex] and a standard deviation of [latex]1[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1799\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1799&theme=lumen&iframe_resize_id=ohm1799&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\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=\"ohm975\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=975&theme=lumen&iframe_resize_id=ohm975&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":15,"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\/1284"}],"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\/1284\/revisions"}],"predecessor-version":[{"id":6787,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1284\/revisions\/6787"}],"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\/1284\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1284"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1284"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1284"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}