{"id":1219,"date":"2021-07-01T14:31:08","date_gmt":"2021-07-01T18:31:08","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=1219"},"modified":"2024-02-08T14:42:06","modified_gmt":"2024-02-08T19:42:06","slug":"12-3-f-distribution","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/12-3-f-distribution\/","title":{"raw":"12.3 F Distribution","rendered":"12.3 F Distribution"},"content":{"raw":"In a two-sample [latex]t[\/latex]\u00a0test, we use the [latex]t[\/latex]\u00a0distribution to calculate the <em>P-<\/em>value and find the critical value. ANOVA relies on the F distribution whose density curve is unimodal, right skewed, and has two degrees of freedom. The ratio of MSTR over MSE follows an F distribution with degrees of freedom [latex]df_n = k-1[\/latex] and [latex]df_d = n-k[\/latex].\r\n\r\nThe figure below shows the density curve of several F distributions with different degrees of freedom.<a id=\"retfig12.3\"><\/a>\r\n\r\n[caption id=\"attachment_1223\" align=\"aligncenter\" width=\"410\"]<img class=\"wp-image-1223 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008.png\" alt=\"Several F-distributions of varying degrees of freedom. It bears some resemblance to the chi-squared graph. Image description available.\" width=\"410\" height=\"382\" \/> <strong>Figure 12.3<\/strong>: F Density Curves. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.3\">Image Description (See Appendix D Figure 12.3)<\/a>][\/caption]The properties of F-density curves are as follows:\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Fact: Properties of F Density Curve<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Total area under the curve=1.<\/li>\r\n \t<li>The entire curve is above the horizontal axis.<\/li>\r\n \t<li>Right skewed.<\/li>\r\n \t<li>Horizontal axis spans from 0 to [latex]+ \\infty[\/latex]<\/li>\r\n \t<li>The two degrees of freedom [latex]df_n[\/latex]\u00a0and [latex]df_d[\/latex]\u00a0control the overall shape.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<span style=\"text-align: initial; font-size: 1em;\">Similar to the [latex]t[\/latex] score table (<\/span><a style=\"text-align: initial; font-size: 1em;\" href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/08\/t_table_more_crop.png\" target=\"_blank\" rel=\"noopener\">Table IV<\/a><span style=\"text-align: initial; font-size: 1em;\">), the F Table (see Table VI below) gives scores that correspond to right-tailed probabilities. For given numerator degrees of freedom [latex](df_n)[\/latex] and denominator degrees of freedom [latex](df_d)[\/latex], the F-score [latex]F_{\\alpha}[\/latex] is the value with an area [latex]\\alpha[\/latex] to its right. For example, for [latex]df_n =2 , df_d = 3, F_{0.1}=5.462, F_{0.01}=30.82[\/latex], and [latex]F_{0.005}=49.8[\/latex].<\/span>\r\n\r\n<strong>Table 12.3<\/strong>: Part of the F Table (Table VI)<a id=\"rettab12.1\"><\/a>\r\n\r\n[caption id=\"attachment_3095\" align=\"aligncenter\" width=\"5173\"]<img class=\"wp-image-3095 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table.png\" alt=\"Part of the F-table. Image description available.\" width=\"5173\" height=\"4845\" \/> [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#tab12.1\">Image Description (See Appendix D Table 12.3)<\/a>][\/caption]","rendered":"<p>In a two-sample [latex]t[\/latex]\u00a0test, we use the [latex]t[\/latex]\u00a0distribution to calculate the <em>P-<\/em>value and find the critical value. ANOVA relies on the F distribution whose density curve is unimodal, right skewed, and has two degrees of freedom. The ratio of MSTR over MSE follows an F distribution with degrees of freedom [latex]df_n = k-1[\/latex] and [latex]df_d = n-k[\/latex].<\/p>\n<p>The figure below shows the density curve of several F distributions with different degrees of freedom.<a id=\"retfig12.3\"><\/a><\/p>\n<figure id=\"attachment_1223\" aria-describedby=\"caption-attachment-1223\" style=\"width: 410px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1223 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008.png\" alt=\"Several F-distributions of varying degrees of freedom. It bears some resemblance to the chi-squared graph. Image description available.\" width=\"410\" height=\"382\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008.png 410w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008-300x280.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008-65x61.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008-225x210.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/M12-02-OpeningExample_clip_image008-350x326.png 350w\" sizes=\"auto, (max-width: 410px) 100vw, 410px\" \/><figcaption id=\"caption-attachment-1223\" class=\"wp-caption-text\"><strong>Figure 12.3<\/strong>: F Density Curves. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.3\">Image Description (See Appendix D Figure 12.3)<\/a>]<\/figcaption><\/figure>\n<p>The properties of F-density curves are as follows:<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Fact: Properties of F Density Curve<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Total area under the curve=1.<\/li>\n<li>The entire curve is above the horizontal axis.<\/li>\n<li>Right skewed.<\/li>\n<li>Horizontal axis spans from 0 to [latex]+ \\infty[\/latex]<\/li>\n<li>The two degrees of freedom [latex]df_n[\/latex]\u00a0and [latex]df_d[\/latex]\u00a0control the overall shape.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><span style=\"text-align: initial; font-size: 1em;\">Similar to the [latex]t[\/latex] score table (<\/span><a style=\"text-align: initial; font-size: 1em;\" href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/08\/t_table_more_crop.png\" target=\"_blank\" rel=\"noopener\">Table IV<\/a><span style=\"text-align: initial; font-size: 1em;\">), the F Table (see Table VI below) gives scores that correspond to right-tailed probabilities. For given numerator degrees of freedom [latex](df_n)[\/latex] and denominator degrees of freedom [latex](df_d)[\/latex], the F-score [latex]F_{\\alpha}[\/latex] is the value with an area [latex]\\alpha[\/latex] to its right. For example, for [latex]df_n =2 , df_d = 3, F_{0.1}=5.462, F_{0.01}=30.82[\/latex], and [latex]F_{0.005}=49.8[\/latex].<\/span><\/p>\n<p><strong>Table 12.3<\/strong>: Part of the F Table (Table VI)<a id=\"rettab12.1\"><\/a><\/p>\n<figure id=\"attachment_3095\" aria-describedby=\"caption-attachment-3095\" style=\"width: 5173px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3095 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table.png\" alt=\"Part of the F-table. Image description available.\" width=\"5173\" height=\"4845\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table.png 5173w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-300x281.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-1024x959.png 1024w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-768x719.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-1536x1439.png 1536w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-2048x1918.png 2048w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-65x61.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-225x211.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/F_table-350x328.png 350w\" sizes=\"auto, (max-width: 5173px) 100vw, 5173px\" \/><figcaption id=\"caption-attachment-3095\" class=\"wp-caption-text\">[<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#tab12.1\">Image Description (See Appendix D Table 12.3)<\/a>]<\/figcaption><\/figure>\n","protected":false},"author":19,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1219","chapter","type-chapter","status-publish","hentry"],"part":1189,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":26,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1219\/revisions"}],"predecessor-version":[{"id":5203,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1219\/revisions\/5203"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/1189"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1219\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1219"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=1219"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1219"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}