{"id":1287,"date":"2023-06-22T02:13:19","date_gmt":"2023-06-22T02:13:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/t-distribution-apply-it-2\/"},"modified":"2025-05-16T03:59:31","modified_gmt":"2025-05-16T03:59:31","slug":"t-distribution-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/t-distribution-apply-it-2\/","title":{"raw":"t-distribution: Learn It 4","rendered":"t-distribution: Learn It 4"},"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>[latex]z[\/latex]-statistic vs. [latex]t[\/latex]-statistic<\/h2>\r\n<section class=\"textbox recall\">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>.\r\n\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<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1797[\/ohm2_question]<\/section>\r\n<p>Suppose we do not know the value of the population standard deviation, [latex]\\sigma[\/latex]. If we want to calculate the value of a standardized sample mean, we need to know [latex]\\sigma[\/latex]. So, instead of [latex]\\sigma[\/latex], let\u2019s substitute in our best estimate for [latex]\\sigma[\/latex]: the sample standard deviation, [latex]s[\/latex].<\/p>\r\n<p>Recall that an estimate of the standard deviation of a statistic is called the <strong>standard error<\/strong> of that statistic. Since we estimate the standard deviation of the sample mean [latex]\\dfrac{\\sigma}{\\sqrt{n}}[\/latex] by substituting in [latex]s[\/latex] for [latex]\\sigma[\/latex], the <strong>standard error<\/strong> of the sample mean is [latex]SE(\\bar{x})=\\dfrac{s}{\\sqrt{n}}[\/latex].<\/p>\r\n<section class=\"textbox recall\">When we estimate [latex]\\sigma[\/latex] using the sample standard deviation, [latex]s[\/latex], we use the [latex]t[\/latex]-<strong>statistic:<\/strong>\r\n<p style=\"text-align: center;\">[latex]t=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. error of } \\bar{x}'s} [\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1798[\/ohm2_question]<\/section>\r\n<p>The distribution of [latex]z[\/latex]-scores is the standard normal curve, with a mean of 0 and a standard deviation of 1. The distribution of [latex]t[\/latex]-scores depends on the sample size, [latex]n[\/latex]. There is a different [latex]t[\/latex]-model for every [latex]n[\/latex]. Instead of referring to [latex]n[\/latex] to specify which [latex]t[\/latex]-model to use, we refer to the <strong>degrees of freedom<\/strong>, or [latex]df[\/latex] for short. The number of degrees of freedom is 1 less than the sample size. That is, [latex]df = n \u2013 1[\/latex].<\/p>\r\n<p>Let's see how the degree of freedom impacts the [latex]t[\/latex]-distribution curve.<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/tdist\/ \" 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\/tdist\/\" 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]1915[\/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>[latex]z[\/latex]-statistic vs. [latex]t[\/latex]-statistic<\/h2>\n<section class=\"textbox recall\">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>.<\/p>\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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1797\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1797&theme=lumen&iframe_resize_id=ohm1797&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Suppose we do not know the value of the population standard deviation, [latex]\\sigma[\/latex]. If we want to calculate the value of a standardized sample mean, we need to know [latex]\\sigma[\/latex]. So, instead of [latex]\\sigma[\/latex], let\u2019s substitute in our best estimate for [latex]\\sigma[\/latex]: the sample standard deviation, [latex]s[\/latex].<\/p>\n<p>Recall that an estimate of the standard deviation of a statistic is called the <strong>standard error<\/strong> of that statistic. Since we estimate the standard deviation of the sample mean [latex]\\dfrac{\\sigma}{\\sqrt{n}}[\/latex] by substituting in [latex]s[\/latex] for [latex]\\sigma[\/latex], the <strong>standard error<\/strong> of the sample mean is [latex]SE(\\bar{x})=\\dfrac{s}{\\sqrt{n}}[\/latex].<\/p>\n<section class=\"textbox recall\">When we estimate [latex]\\sigma[\/latex] using the sample standard deviation, [latex]s[\/latex], we use the [latex]t[\/latex]&#8211;<strong>statistic:<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]t=\\dfrac{\\bar{x}-[\\text{mean of } \\bar{x}'s]}{\\text{std. error of } \\bar{x}'s}[\/latex] [latex]= \\dfrac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1798\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1798&theme=lumen&iframe_resize_id=ohm1798&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>The distribution of [latex]z[\/latex]-scores is the standard normal curve, with a mean of 0 and a standard deviation of 1. The distribution of [latex]t[\/latex]-scores depends on the sample size, [latex]n[\/latex]. There is a different [latex]t[\/latex]-model for every [latex]n[\/latex]. Instead of referring to [latex]n[\/latex] to specify which [latex]t[\/latex]-model to use, we refer to the <strong>degrees of freedom<\/strong>, or [latex]df[\/latex] for short. The number of degrees of freedom is 1 less than the sample size. That is, [latex]df = n \u2013 1[\/latex].<\/p>\n<p>Let&#8217;s see how the degree of freedom impacts the [latex]t[\/latex]-distribution curve.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/tdist\/\" 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\/tdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1915\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1915&theme=lumen&iframe_resize_id=ohm1915&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":17,"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\/1287"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions"}],"predecessor-version":[{"id":6789,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions\/6789"}],"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\/1287\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1287"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1287"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1287"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}