{"id":1288,"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-3\/"},"modified":"2024-01-18T23:47:41","modified_gmt":"2024-01-18T23:47:41","slug":"t-distribution-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/t-distribution-apply-it-3\/","title":{"raw":"t-distribution: Apply It 2","rendered":"t-distribution: 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;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<section class=\"textbox recall\">When taking many, many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex], the [latex]t[\/latex]-statistic:\r\n\r\n<p style=\"text-align: center;\">[latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{SE(\\stackrel{\u00af}{x})} = \\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\r\n<p>will follow a [latex]t[\/latex]-distribution with<strong> [latex]n-1[\/latex] degrees of freedom<\/strong> if the population distribution is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n \\ge 30[\/latex]).<\/p>\r\n<\/section>\r\n<p>Since the [latex]t[\/latex]-statistic exhibits more sampling variability than the [latex]z[\/latex]-statistic, its distribution has slightly more variability than a standard normal distribution. However, as the sample size increases, there is less sampling variability associated with the standard error of the sample mean, so its distribution gets closer to a standard normal distribution.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]970[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]971[\/ohm2_question]<\/section>\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>","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<section class=\"textbox recall\">When taking many, many random samples of size [latex]n[\/latex] from a population distribution with mean [latex]\\mu[\/latex] and standard deviation [latex]\\sigma[\/latex], the [latex]t[\/latex]-statistic:<\/p>\n<p style=\"text-align: center;\">[latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{SE(\\stackrel{\u00af}{x})} = \\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\n<p>will follow a [latex]t[\/latex]-distribution with<strong> [latex]n-1[\/latex] degrees of freedom<\/strong> if the population distribution is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n \\ge 30[\/latex]).<\/p>\n<\/section>\n<p>Since the [latex]t[\/latex]-statistic exhibits more sampling variability than the [latex]z[\/latex]-statistic, its distribution has slightly more variability than a standard normal distribution. However, as the sample size increases, there is less sampling variability associated with the standard error of the sample mean, so its distribution gets closer to a standard normal distribution.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm970\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=970&theme=lumen&iframe_resize_id=ohm970&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm971\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=971&theme=lumen&iframe_resize_id=ohm971&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\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","protected":false},"author":8,"menu_order":19,"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\/1288"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1288\/revisions"}],"predecessor-version":[{"id":5152,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1288\/revisions\/5152"}],"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\/1288\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1288"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1288"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1288"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}