{"id":1226,"date":"2021-07-01T15:01:38","date_gmt":"2021-07-01T19:01:38","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=1226"},"modified":"2024-02-08T14:43:47","modified_gmt":"2024-02-08T19:43:47","slug":"12-4-one-way-anova-f-test","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/12-4-one-way-anova-f-test\/","title":{"raw":"12.4 One-Way ANOVA F Test","rendered":"12.4 One-Way ANOVA F Test"},"content":{"raw":"The assumptions and steps of a one-way ANOVA F test are as follows.\r\n<div class=\"textbox\">\r\n\r\n<strong>Assumptions<\/strong>:\r\n<ul>\r\n \t<li>Normal populations: the variable of interest is normally distributed for each population.<\/li>\r\n \t<li>Equal variances: the variance of the variable of interest is the same for all populations.<\/li>\r\n \t<li>Independent samples: the samples from different populations are independent of one another.<\/li>\r\n \t<li>Simple random samples: the samples taken from the [latex]k[\/latex]\u00a0populations are simple random samples.<\/li>\r\n<\/ul>\r\n<strong>Steps<\/strong>:\r\n<ol>\r\n \t<li>Set up the hypotheses:\r\n[latex]\\begin{align*} H_0 &amp;: \\mu_1 = \\mu_2 = \\dots = \\mu_k\\\\ H_a&amp;: \\text{Not all means are equal.}\\end{align*}[\/latex]<\/li>\r\n \t<li>State the significance level [latex]\\alpha[\/latex].<\/li>\r\n \t<li>Calculate the sums of squares SST, SSTR, SSE and the mean squares MSTR, MSE. Find the test statistic, [latex]F_o[\/latex], and show the results in an ANOVA table:\r\n<div align=\"center\">\r\n<table class=\"first-col-border\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr class=\"border-bottom\">\r\n<th style=\"width: 105px;\" scope=\"col\" valign=\"top\">Source<\/th>\r\n<th style=\"text-align: center; width: 112.722px;\" scope=\"col\" valign=\"top\">\r\n<div align=\"center\">[latex]df[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center; width: 135px;\" scope=\"col\" valign=\"top\">\r\n<div align=\"center\">[latex]SS[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center; width: 183px;\" scope=\"col\" valign=\"top\">\r\n<div align=\"center\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\r\n<div align=\"center\">F-statistic<\/div><\/th>\r\n<th style=\"text-align: center; width: 121px;\" scope=\"col\" valign=\"top\">\r\n<div align=\"center\">p-value<\/div><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Treatment<\/th>\r\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\r\n<div align=\"center\">[latex]k-1[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]SSTR[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]MSTR = \\frac{SSTR}{k-1}[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]F_o = \\frac{MSTR}{MSE}[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\">\r\n<div align=\"center\">[latex]P(F \\geq F_o)[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Error<\/th>\r\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\r\n<div align=\"center\">[latex]n-k[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]SSE[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]MSE = \\frac{SSE}{n-k}[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\"><\/td>\r\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Total<\/th>\r\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\r\n<div align=\"center\">[latex]n-1[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\r\n<div align=\"center\">[latex]SST[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\"><\/td>\r\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\"><\/td>\r\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n \t<li>Find the P-value <strong>or<\/strong> rejection region based on the F density curve with degrees of freedom [latex]df_n = k-1, df_d = n-k[\/latex].\r\n<div align=\"center\">\r\n<table class=\"first-col-border\" style=\"width: 100%;\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 131.9375px;\" scope=\"row\" valign=\"top\">P-value<\/th>\r\n<td style=\"width: 454.203125px; text-align: left;\" valign=\"top\">[latex]P(F \\geq F_o)[\/latex] the area to the right of [latex]F_o[\/latex]\u00a0under the curve<\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 131.9375px;\" scope=\"row\" valign=\"top\">Rejection region<\/th>\r\n<td style=\"width: 454.203125px; text-align: left;\" valign=\"top\">[latex]F \\geq F_{\\alpha}[\/latex] the region to the right of the critical value [latex]F_{\\alpha}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n \t<li>Reject the null [latex]H_0[\/latex]\u00a0if P-value [latex]\\leq \\alpha[\/latex]\u00a0or [latex]F_o[\/latex]\u00a0falls in the rejection region.<\/li>\r\n \t<li>Conclusion.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: One-Way ANOVA F Test on Download Time<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe following ANOVA table corresponds to the download time example. Use the information in\u00a0this ANOVA table to test at the 1% significance level whether there is a significant difference between the mean download times at 7 a.m., 5 p.m., and 12 a.m.\r\n<p style=\"text-align: center;\"><strong>Table 12.4<\/strong>: ANOVA Table of Download Time Example<\/p>\r\n\r\n<div align=\"center\">\r\n<table class=\"first-col-border\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr class=\"border-bottom\">\r\n<th scope=\"col\" valign=\"top\" width=\"95\">Source<\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"76\">\r\n<div align=\"center\">[latex]df[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"107\">\r\n<div align=\"center\">[latex]SS[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"129\">\r\n<div align=\"center\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">F-statistic<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">P-value<\/div><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Time of day<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]2[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]204641[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\">\r\n<div align=\"center\">[latex]102320[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">[latex]F_o = 46.03[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">[latex]&lt; 0.0001[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Error<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]45[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]100020[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\">\r\n<div align=\"center\">[latex]2223[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Total<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]47[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]304661[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThe side-by-side histograms and boxplots of all three groups can be found below. Both the side-by-side histograms and boxplots show that the downloading time at 7 AM, 5 PM and 12 AM is not normally distributed, since the histograms are not bell-shaped and the boxplots are not symmetric. The side-by-side boxplots show that the median downloading time of 7 AM, 5 PM, and 12 AM are around 90, 260, and 200 minutes respectively.<a id=\"retfig12.4\"><\/a>\r\n\r\n[caption id=\"attachment_2917\" align=\"aligncenter\" width=\"600\"]<img class=\"wp-image-2917\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download.png\" alt=\"Three histograms of download time for 7 am 5 pm and 12 am are shown. Below them are their corresponding box plots. Image description available.\" width=\"600\" height=\"600\" \/> <strong>Figure 12.4<\/strong>: Side-by-side Histograms and Boxplots of Time of Day. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.4\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.4\">(See Appendix D Figure 12.4)<\/a>][\/caption]<strong>Steps to conduct a one-way ANOVA F test<\/strong>:\r\n<ol>\r\n \t<li>Hypotheses\r\n<p align=\"left\">[latex]H_0: \\mu_1 = \\mu_2 = \\mu_3[\/latex]\r\n[latex]H_a: \\text{Not all means are equal}[\/latex].<\/p>\r\n<\/li>\r\n \t<li>The significance level is\u00a0[latex]\\alpha = 0.01[\/latex].<\/li>\r\n \t<li>The test statistic is\u00a0[latex]F_o = 46.03[\/latex]\u00a0with [latex]df_n = k-1 = 3-1 =2, df_d = n-k = 48 -3 =45[\/latex].<\/li>\r\n \t<li>P-value [latex]P(F \\geq F_o) = P(F \\geq 46.03) &lt; 0.0001[\/latex] (given in the ANOVA table)<\/li>\r\n \t<li>Reject [latex]H_0[\/latex], since p-value [latex]&lt; 0.0001 &lt; 0.01 (\\alpha)[\/latex].<\/li>\r\n \t<li>Conclusion: At the 1% significance level, we have sufficient evidence that there is a significant difference among the mean download times at 7 a.m., 5 p.m., and 12 a.m.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div style=\"height: 55px; margin-top: 5px;\"><img class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" \/><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercises: One-Way ANOVA F Test<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMany studies have suggested that there is an association between exercise and healthy bones. One study examined the effect of jumping on the bone density of growing rats. There are three treatments: a control with no jumping, a low-jump condition (the jump height was 30 centimetres), and a high-jump condition (60 centimetres). After eight weeks of 10 jumps per day, five days per week, the bone density of the rats in milligrams per cubic centimetre (mg\/cm<sup>3<\/sup>) was measured. The data are given in the following table.\r\n<p style=\"text-align: center;\"><strong>Table 12.5<\/strong>: Bone Density for Three Treatments<\/p>\r\n\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 150px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">611<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">635<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">650<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">621<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">605<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">622<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">614<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">638<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">626<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">593<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">594<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">626<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">593<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">699<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">631<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">653<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">632<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">622<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">600<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">631<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">643<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">554<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">588<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">674<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">603<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">607<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">643<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">569<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">596<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\r\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">650<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe side-by-side histograms and boxplots are shown as follows:<a id=\"retfig12.5\"><\/a>\r\n\r\n[caption id=\"attachment_2921\" align=\"aligncenter\" width=\"600\"]<img class=\"wp-image-2921\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity.png\" alt=\"Three histograms of bone density for control, high jump and low jump. Their corresponding box plots are shown below. Image description available.\" width=\"600\" height=\"600\" \/> <strong>Figure 12.5<\/strong>: Side-by-side Histograms and Boxplots of Groups. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.5\">Image Description (See Appendix D Figure 12.5)<\/a>][\/caption]Given the ANOVA table, test at the 5% significance level whether jumping strengthens the bones of rats.\r\n<p style=\"text-align: center;\"><strong>Table 12.6<\/strong>: ANOVA Table of Bone Density Exercise<\/p>\r\n\r\n<div align=\"center\">\r\n<table class=\"no-lines first-col-border\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr class=\"border-bottom\">\r\n<th scope=\"col\" valign=\"top\" width=\"95\">Source<\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"76\">\r\n<div align=\"center\">[latex]df[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"107\">\r\n<div align=\"center\">[latex]SS[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"129\">\r\n<div align=\"center\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">F-statistic<\/div><\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">P-value<\/div><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Group<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]2[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]7114[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\">\r\n<div align=\"center\">[latex]3557[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">[latex]F_o = 5.08[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\">\r\n<div align=\"center\">[latex] 0.0133[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Error<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]27[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]18879[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\">\r\n<div align=\"center\">[latex]699[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th scope=\"row\" valign=\"top\" width=\"123\">Total<\/th>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"66\">\r\n<div align=\"center\">[latex]29[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"95\">\r\n<div align=\"center\">[latex]25993[\/latex]<\/div><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"142\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<td style=\"text-align: center;\" valign=\"top\" width=\"123\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<details><summary>Show\/Hide Answer<\/summary><strong>Answers:<\/strong>\r\n\r\n<strong>Steps to conduct a one-way ANOVA F test<\/strong>:\r\n<ol>\r\n \t<li>Hypotheses\r\n[latex]H_0: \\mu_1 = \\mu_2 = \\mu_3[\/latex]\r\n[latex]H_a: \\text{Not all means are equal}[\/latex].<\/li>\r\n \t<li>The significance level is [latex]\\alpha = 0.05[\/latex].<\/li>\r\n \t<li>The test statistic is [latex]F_o = 5.08[\/latex]\u00a0with [latex]df_n = k-1 = 3-1 = 2, df_d = n-k = 30-3 = 27[\/latex].<\/li>\r\n \t<li>P-value [latex]= P(F \\geq F_o) = P(F \\geq 5.08) &lt; 0.0133[\/latex] (given in the ANOVA table)<\/li>\r\n \t<li>Reject [latex]H_0[\/latex], since p\u2013value [latex]= 0.0133 &lt; 0.05 (\\alpha)[\/latex].<\/li>\r\n \t<li>Conclusion: At the 5% significance level, we have sufficient evidence that the mean bone densities are different in the three treatment groups. Since the rats in the \"high jump\" group has the largest mean bone density, followed by the \"low jump\" group, we can conclude that jumping strengthens the bones of rats.<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>","rendered":"<p>The assumptions and steps of a one-way ANOVA F test are as follows.<\/p>\n<div class=\"textbox\">\n<p><strong>Assumptions<\/strong>:<\/p>\n<ul>\n<li>Normal populations: the variable of interest is normally distributed for each population.<\/li>\n<li>Equal variances: the variance of the variable of interest is the same for all populations.<\/li>\n<li>Independent samples: the samples from different populations are independent of one another.<\/li>\n<li>Simple random samples: the samples taken from the [latex]k[\/latex]\u00a0populations are simple random samples.<\/li>\n<\/ul>\n<p><strong>Steps<\/strong>:<\/p>\n<ol>\n<li>Set up the hypotheses:<br \/>\n[latex]\\begin{align*} H_0 &: \\mu_1 = \\mu_2 = \\dots = \\mu_k\\\\ H_a&: \\text{Not all means are equal.}\\end{align*}[\/latex]<\/li>\n<li>State the significance level [latex]\\alpha[\/latex].<\/li>\n<li>Calculate the sums of squares SST, SSTR, SSE and the mean squares MSTR, MSE. Find the test statistic, [latex]F_o[\/latex], and show the results in an ANOVA table:\n<div style=\"margin: auto;\">\n<table class=\"first-col-border\" cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr class=\"border-bottom\">\n<th style=\"width: 105px;\" scope=\"col\" valign=\"top\">Source<\/th>\n<th style=\"text-align: center; width: 112.722px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]df[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 135px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SS[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 183px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">F-statistic<\/div>\n<\/th>\n<th style=\"text-align: center; width: 121px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">p-value<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Treatment<\/th>\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]k-1[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SSTR[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]MSTR = \\frac{SSTR}{k-1}[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]F_o = \\frac{MSTR}{MSE}[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]P(F \\geq F_o)[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Error<\/th>\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]n-k[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SSE[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]MSE = \\frac{SSE}{n-k}[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\"><\/td>\n<\/tr>\n<tr>\n<th style=\"width: 105px;\" scope=\"row\" valign=\"top\">Total<\/th>\n<td style=\"text-align: center; width: 113.167px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]n-1[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 135.889px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SST[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 183.889px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123.889px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 121.444px;\" valign=\"top\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n<li>Find the P-value <strong>or<\/strong> rejection region based on the F density curve with degrees of freedom [latex]df_n = k-1, df_d = n-k[\/latex].\n<div style=\"margin: auto;\">\n<table class=\"first-col-border\" style=\"width: 100%; border-spacing: 0px;\" cellpadding=\"0\">\n<tbody>\n<tr>\n<th style=\"width: 131.9375px;\" scope=\"row\" valign=\"top\">P-value<\/th>\n<td style=\"width: 454.203125px; text-align: left;\" valign=\"top\">[latex]P(F \\geq F_o)[\/latex] the area to the right of [latex]F_o[\/latex]\u00a0under the curve<\/td>\n<\/tr>\n<tr>\n<th style=\"width: 131.9375px;\" scope=\"row\" valign=\"top\">Rejection region<\/th>\n<td style=\"width: 454.203125px; text-align: left;\" valign=\"top\">[latex]F \\geq F_{\\alpha}[\/latex] the region to the right of the critical value [latex]F_{\\alpha}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n<li>Reject the null [latex]H_0[\/latex]\u00a0if P-value [latex]\\leq \\alpha[\/latex]\u00a0or [latex]F_o[\/latex]\u00a0falls in the rejection region.<\/li>\n<li>Conclusion.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: One-Way ANOVA F Test on Download Time<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The following ANOVA table corresponds to the download time example. Use the information in\u00a0this ANOVA table to test at the 1% significance level whether there is a significant difference between the mean download times at 7 a.m., 5 p.m., and 12 a.m.<\/p>\n<p style=\"text-align: center;\"><strong>Table 12.4<\/strong>: ANOVA Table of Download Time Example<\/p>\n<div style=\"margin: auto;\">\n<table class=\"first-col-border\" cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr class=\"border-bottom\">\n<th scope=\"col\" valign=\"top\" style=\"width: 95px;\">Source<\/th>\n<th style=\"text-align: center; width: 76px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]df[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 107px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SS[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 129px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">F-statistic<\/div>\n<\/th>\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">P-value<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Time of day<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]2[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]204641[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]102320[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]F_o = 46.03[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]< 0.0001[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Error<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]45[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]100020[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]2223[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<\/tr>\n<tr>\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Total<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]47[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]304661[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>The side-by-side histograms and boxplots of all three groups can be found below. Both the side-by-side histograms and boxplots show that the downloading time at 7 AM, 5 PM and 12 AM is not normally distributed, since the histograms are not bell-shaped and the boxplots are not symmetric. The side-by-side boxplots show that the median downloading time of 7 AM, 5 PM, and 12 AM are around 90, 260, and 200 minutes respectively.<a id=\"retfig12.4\"><\/a><\/p>\n<figure id=\"attachment_2917\" aria-describedby=\"caption-attachment-2917\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2917\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download.png\" alt=\"Three histograms of download time for 7 am 5 pm and 12 am are shown. Below them are their corresponding box plots. Image description available.\" width=\"600\" height=\"600\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download.png 480w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_download-350x350.png 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-2917\" class=\"wp-caption-text\"><strong>Figure 12.4<\/strong>: Side-by-side Histograms and Boxplots of Time of Day. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.4\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.4\">(See Appendix D Figure 12.4)<\/a>]<\/figcaption><\/figure>\n<p><strong>Steps to conduct a one-way ANOVA F test<\/strong>:<\/p>\n<ol>\n<li>Hypotheses\n<p style=\"text-align: left;\">[latex]H_0: \\mu_1 = \\mu_2 = \\mu_3[\/latex]<br \/>\n[latex]H_a: \\text{Not all means are equal}[\/latex].<\/p>\n<\/li>\n<li>The significance level is\u00a0[latex]\\alpha = 0.01[\/latex].<\/li>\n<li>The test statistic is\u00a0[latex]F_o = 46.03[\/latex]\u00a0with [latex]df_n = k-1 = 3-1 =2, df_d = n-k = 48 -3 =45[\/latex].<\/li>\n<li>P-value [latex]P(F \\geq F_o) = P(F \\geq 46.03) < 0.0001[\/latex] (given in the ANOVA table)<\/li>\n<li>Reject [latex]H_0[\/latex], since p-value [latex]< 0.0001 < 0.01 (\\alpha)[\/latex].<\/li>\n<li>Conclusion: At the 1% significance level, we have sufficient evidence that there is a significant difference among the mean download times at 7 a.m., 5 p.m., and 12 a.m.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"height: 55px; margin-top: 5px;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png 250w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-65x13.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-225x45.png 225w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercises: One-Way ANOVA F Test<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Many studies have suggested that there is an association between exercise and healthy bones. One study examined the effect of jumping on the bone density of growing rats. There are three treatments: a control with no jumping, a low-jump condition (the jump height was 30 centimetres), and a high-jump condition (60 centimetres). After eight weeks of 10 jumps per day, five days per week, the bone density of the rats in milligrams per cubic centimetre (mg\/cm<sup>3<\/sup>) was measured. The data are given in the following table.<\/p>\n<p style=\"text-align: center;\"><strong>Table 12.5<\/strong>: Bone Density for Three Treatments<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 150px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Group<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\"><strong>Bone density<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">611<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">635<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">650<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">621<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">605<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">622<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">614<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">638<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">626<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">593<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">594<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">626<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">593<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">699<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">631<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">653<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">632<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">622<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">600<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">631<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">643<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">554<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">588<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">674<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">603<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">607<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">643<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Control<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">569<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">Low jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">596<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">High jump<\/td>\n<td style=\"width: 16.6667%; height: 15px; text-align: center;\">650<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The side-by-side histograms and boxplots are shown as follows:<a id=\"retfig12.5\"><\/a><\/p>\n<figure id=\"attachment_2921\" aria-describedby=\"caption-attachment-2921\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2921\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity.png\" alt=\"Three histograms of bone density for control, high jump and low jump. Their corresponding box plots are shown below. Image description available.\" width=\"600\" height=\"600\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity.png 480w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/07\/ANOVA_bonedensity-350x350.png 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-2921\" class=\"wp-caption-text\"><strong>Figure 12.5<\/strong>: Side-by-side Histograms and Boxplots of Groups. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig12.5\">Image Description (See Appendix D Figure 12.5)<\/a>]<\/figcaption><\/figure>\n<p>Given the ANOVA table, test at the 5% significance level whether jumping strengthens the bones of rats.<\/p>\n<p style=\"text-align: center;\"><strong>Table 12.6<\/strong>: ANOVA Table of Bone Density Exercise<\/p>\n<div style=\"margin: auto;\">\n<table class=\"no-lines first-col-border\" cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr class=\"border-bottom\">\n<th scope=\"col\" valign=\"top\" style=\"width: 95px;\">Source<\/th>\n<th style=\"text-align: center; width: 76px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]df[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 107px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]SS[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 129px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]MS = \\frac{SS}{df}[\/latex]<\/div>\n<\/th>\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">F-statistic<\/div>\n<\/th>\n<th style=\"text-align: center; width: 123px;\" scope=\"col\" valign=\"top\">\n<div style=\"margin: auto;\">P-value<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Group<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]2[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]7114[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]3557[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]F_o = 5.08[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]0.0133[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Error<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]27[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]18879[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]699[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<\/tr>\n<tr>\n<th scope=\"row\" valign=\"top\" style=\"width: 123px;\">Total<\/th>\n<td style=\"text-align: center; width: 66px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]29[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 95px;\" valign=\"top\">\n<div style=\"margin: auto;\">[latex]25993[\/latex]<\/div>\n<\/td>\n<td style=\"text-align: center; width: 142px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<td style=\"text-align: center; width: 123px;\" valign=\"top\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<details>\n<summary>Show\/Hide Answer<\/summary>\n<p><strong>Answers:<\/strong><\/p>\n<p><strong>Steps to conduct a one-way ANOVA F test<\/strong>:<\/p>\n<ol>\n<li>Hypotheses<br \/>\n[latex]H_0: \\mu_1 = \\mu_2 = \\mu_3[\/latex]<br \/>\n[latex]H_a: \\text{Not all means are equal}[\/latex].<\/li>\n<li>The significance level is [latex]\\alpha = 0.05[\/latex].<\/li>\n<li>The test statistic is [latex]F_o = 5.08[\/latex]\u00a0with [latex]df_n = k-1 = 3-1 = 2, df_d = n-k = 30-3 = 27[\/latex].<\/li>\n<li>P-value [latex]= P(F \\geq F_o) = P(F \\geq 5.08) < 0.0133[\/latex] (given in the ANOVA table)<\/li>\n<li>Reject [latex]H_0[\/latex], since p\u2013value [latex]= 0.0133 < 0.05 (\\alpha)[\/latex].<\/li>\n<li>Conclusion: At the 5% significance level, we have sufficient evidence that the mean bone densities are different in the three treatment groups. Since the rats in the &#8220;high jump&#8221; group has the largest mean bone density, followed by the &#8220;low jump&#8221; group, we can conclude that jumping strengthens the bones of rats.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1226","chapter","type-chapter","status-publish","hentry"],"part":1189,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1226","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":50,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1226\/revisions"}],"predecessor-version":[{"id":5317,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1226\/revisions\/5317"}],"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\/1226\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1226"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=1226"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1226"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1226"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}