{"id":94,"date":"2021-01-12T22:19:48","date_gmt":"2021-01-12T22:19:48","guid":{"rendered":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/goodness-of-fit-test\/"},"modified":"2024-04-28T14:30:28","modified_gmt":"2024-04-28T14:30:28","slug":"goodness-of-fit-test","status":"publish","type":"chapter","link":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/goodness-of-fit-test\/","title":{"rendered":"Goodness-of-Fit Test"},"content":{"raw":"<span style=\"display: none;\">\r\n[latexpage]\r\n<\/span>\r\n<div id=\"d2406e86-3ba5-463c-a86b-7e7e02a03a4e\" class=\"chapter-content-module\" data-type=\"page\" data-cnxml-to-html-ver=\"2.1.0\">\r\n<p id=\"element-112\">In this type of hypothesis test, you determine whether the data <strong>\"fit\"<\/strong> a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. <strong>The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.<\/strong><\/p>\r\n<p id=\"element-908\">The test statistic for a goodness-of-fit test is:<\/p>\r\n$$\\sum_k \\frac{(O-E)^2}{E}$$\r\n\r\nwhere:\r\n<ul id=\"fs-idm37230752\">\r\n \t<li><em data-effect=\"italics\">O<\/em> = <span id=\"term206\" data-type=\"term\">observed values<\/span> (data)<\/li>\r\n \t<li id=\"element-481\"><em data-effect=\"italics\">E<\/em> = <span id=\"term207\" data-type=\"term\">expected values<\/span> (from theory)<\/li>\r\n \t<li id=\"element-402\"><em data-effect=\"italics\">k<\/em> = the number of different data cells or categories<\/li>\r\n<\/ul>\r\n<p id=\"element-512\"><strong>The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true.<\/strong> There are <em data-effect=\"italics\">k<\/em> terms of the form $\\frac{(O-E)^2}{E}$.<\/p>\r\nThe number of degrees of freedom is <em data-effect=\"italics\">df<\/em> = (number of categories \u2013 1) = <em>k<\/em> - 1\r\n<p id=\"element-838\"><strong>The <span id=\"term208\" data-type=\"term\">goodness-of-fit test<\/span> is almost always right-tailed.<\/strong> If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.<\/p>\r\n\r\n<div id=\"eip-587\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\" data-type=\"title\"><span id=\"1\" class=\"os-title-label\" data-type=\"\">Note<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-idm98728224\">The expected value for each cell needs to be at least five in order for you to use this test.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"element-719\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.1<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<p id=\"element-615\">Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism rate follows faculty perception. The faculty expected that a group of 100 students would miss class according to <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl001\">Table 10.1<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl001\" class=\"os-table \">\r\n<table summary=\"Table 10.1 \" data-id=\"M03_Ch03_tbl001\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number of absences per term<\/th>\r\n<th scope=\"col\">Expected number of students<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u20132<\/td>\r\n<td>50<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3\u20135<\/td>\r\n<td>30<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6\u20138<\/td>\r\n<td>12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9\u201311<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12+<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.1<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-952\">A random survey across all mathematics courses was then done to determine the actual number <strong>(observed)<\/strong> of absences in a course. The chart in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl002\">Table 10.2<\/a> displays the results of that survey.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl002\" class=\"os-table \">\r\n<table summary=\"Table 10.2 \" data-id=\"M03_Ch03_tbl002\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number of absences per term<\/th>\r\n<th scope=\"col\">Actual number of students<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u20132<\/td>\r\n<td>35<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3\u20135<\/td>\r\n<td>40<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6\u20138<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9\u201311<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12+<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.2<\/span><\/div>\r\n<\/div>\r\n<p id=\"2\"><span data-type=\"newline\">\r\n<\/span>Determine the null and alternative hypotheses needed to conduct a goodness-of-fit test.<\/p>\r\n<p id=\"element-846\"><strong><em data-effect=\"italics\">H<sub>0<\/sub><\/em>:<\/strong> Student absenteeism <strong>fits<\/strong> faculty perception.<\/p>\r\n<p id=\"element-82\"><span data-type=\"newline\">\r\n<\/span>The alternative hypothesis is the opposite of the null hypothesis.<\/p>\r\n<p id=\"element-421\"><strong><em data-effect=\"italics\">H<sub>1<\/sub><\/em>:<\/strong> Student absenteeism <strong>does not fit<\/strong> faculty perception.<\/p>\r\n\r\n<div id=\"fs-idp16583968\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id6198828\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-451\">a. Can you use the information as it appears in the charts to conduct the goodness-of-fit test?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"id6198852\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\r\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.1<\/span><\/h4>\r\n<div class=\"os-solution-container\">\r\n<p id=\"element-600\">a. <strong>No.<\/strong> Notice that the expected number of absences for the \"12+\" entry is less than five (it is two). Combine that group with the \"9\u201311\" group to create new tables where the number of students for each entry are at least five. The new results are in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl003\">Table 10.3<\/a> and <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl004\">Table 10.4<\/a>.\r\n<span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n<div id=\"M03_Ch03_tbl003\" class=\"os-table \">\r\n<table summary=\"Table 10.3 \" data-id=\"M03_Ch03_tbl003\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\"><strong>Number of absences per term<\/strong><\/th>\r\n<th scope=\"col\"><strong>Expected number of students<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u20132<\/td>\r\n<td>50<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3\u20135<\/td>\r\n<td>30<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6\u20138<\/td>\r\n<td>12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9+<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.3<\/span><\/div>\r\n<\/div>\r\n<div id=\"M03_Ch03_tbl004\" class=\"os-table \">\r\n<table summary=\"Table 10.4 \" data-id=\"M03_Ch03_tbl004\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number of absences per term<\/th>\r\n<th scope=\"col\">Actual number of students<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u20132<\/td>\r\n<td>35<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3\u20135<\/td>\r\n<td>40<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6\u20138<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9+<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.4<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div id=\"element-612\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id6198893\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-496\">b. What is the number of degrees of freedom (<em data-effect=\"italics\">df<\/em>)?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"id6198915\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\r\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.1<\/span><\/h4>\r\n<div class=\"os-solution-container\">\r\n<p id=\"element-601\">b. There are four \"cells\" or categories in each of the new tables.<\/p>\r\n<p id=\"element-6012\"><em data-effect=\"italics\">df<\/em> = number of cells \u2013 1 = 4 \u2013 1 = 3<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idm289894560\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.1<\/span><\/h3>\r\n<\/header><section>\r\n<div id=\"eip-31\" class=\" unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"eip-864\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"eip-988\">A factory manager needs to understand how many products are defective versus how many are produced. The number of expected defects is listed in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl005\">Table 10.5<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl005\" class=\"os-table \">\r\n<table summary=\"Table 10.5 \" data-id=\"M03_Ch03_tbl005\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number produced<\/th>\r\n<th scope=\"col\">Number defective<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u2013100<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>101\u2013200<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>201\u2013300<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>301\u2013400<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>401\u2013500<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.5<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-idm111370528\">A random sample was taken to determine the actual number of defects. <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl006\">Table 10.6<\/a> shows the results of the survey.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl006\" class=\"os-table \">\r\n<table summary=\"Table 10.6 \" data-id=\"M03_Ch03_tbl006\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number produced<\/th>\r\n<th scope=\"col\">Number defective<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0\u2013100<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>101\u2013200<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>201\u2013300<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>301\u2013400<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>401\u2013500<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.6<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-idp74195536\">State the null and alternative hypotheses needed to conduct a goodness-of-fit test, and state the degrees of freedom.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\nIn the following Example 10.2, our solution will use the <strong><em>p<\/em>-value method<\/strong> where we find the <em>p<\/em>-value corresponding to our test statistic and compare it with the significance level given.\r\n\r\nIn Example 10.3, our solution will use the <strong>critical value method<\/strong> where we find the critical value corresponding to the given significance level, and compare the critical value to the test statistic.\r\n\r\nThe <em>p<\/em>-value method and the critical value method are equivalent when determining whether to reject or fail to reject the null hypothesis.\r\n<div id=\"element-962\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.2<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"element-811\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id6215628\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-733\">Employers want to know which days of the week employees are absent in a five-day work week. Most employers would like to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week they had the highest number of employee absences. The results were distributed as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl007\">Table 10.7<\/a>. For the population of employees, do the days for the highest number of absences occur with equal frequencies during a five-day work week? Test at a 5% significance level.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl007\" class=\"os-table \">\r\n<table summary=\"Table 10.7 Day of the Week Employees were Most Absent \" data-id=\"M03_Ch03_tbl007\">\r\n<thead valign=\"top\">\r\n<tr>\r\n<th scope=\"col\"><\/th>\r\n<th scope=\"col\">Monday<\/th>\r\n<th scope=\"col\">Tuesday<\/th>\r\n<th scope=\"col\">Wednesday<\/th>\r\n<th scope=\"col\">Thursday<\/th>\r\n<th scope=\"col\">Friday<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody valign=\"top\">\r\n<tr>\r\n<td>Number of Absences<\/td>\r\n<td>15<\/td>\r\n<td>12<\/td>\r\n<td>9<\/td>\r\n<td>9<\/td>\r\n<td>15<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.7<\/span> <span class=\"os-title\" data-type=\"title\">Day of the Week Employees were Most Absent<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"id6215647\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\r\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.2<\/span><\/h4>\r\n<div class=\"os-solution-container\">\r\n<p id=\"element-13\">The null and alternative hypotheses are:<\/p>\r\n\r\n<ul id=\"element-741\">\r\n \t<li><em data-effect=\"italics\">H<sub>0<\/sub><\/em>: The absent days occur with equal frequencies, that is, they fit a uniform distribution.<\/li>\r\n \t<li><em data-effect=\"italics\">H<sub>1<\/sub><\/em>: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution.<\/li>\r\n<\/ul>\r\n<p id=\"element-71\">If the absent days occur with equal frequencies, then, out of 60 absent days (the total in the sample: 15 + 12 + 9 + 9 + 15 = 60), there would be 12 absences on Monday, 12 on Tuesday, 12 on Wednesday, 12 on Thursday, and 12 on Friday. These numbers are the <strong>expected<\/strong> (<em data-effect=\"italics\">E<\/em>) values. The values in the table are the <strong>observed<\/strong> (<em data-effect=\"italics\">O<\/em>) values or data.<\/p>\r\n<p id=\"element-489\">This time, calculate the <em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> test statistic by hand. Make a chart with the following headings and fill in the columns:<\/p>\r\n\r\n<ul id=\"element-120\">\r\n \t<li>Expected (<em data-effect=\"italics\">E<\/em>) values (12, 12, 12, 12, 12)<\/li>\r\n \t<li>Observed (<em data-effect=\"italics\">O<\/em>) values (15, 12, 9, 9, 15)<\/li>\r\n \t<li>(<em data-effect=\"italics\">O<\/em> \u2013 <em data-effect=\"italics\">E<\/em>)<\/li>\r\n \t<li>(<em data-effect=\"italics\">O<\/em> \u2013 <em data-effect=\"italics\">E<\/em>)<sup>2<\/sup><\/li>\r\n \t<li>$\\frac{(O-E)^2}{E}$<\/li>\r\n<\/ul>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 31px;\">\r\n<th style=\"width: 20%;\">Day of Absences<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$O$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$E$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$O - E$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$(O - E)^2$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$\\frac{(O - E)^2}{E}$<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 10%;\">Monday<\/td>\r\n<td style=\"width: 10%; height: 15px;\">15<\/td>\r\n<td style=\"width: 20%; height: 15px;\">12<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">3<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 10%;\">Tuesday<\/td>\r\n<td style=\"width: 10%; height: 15px;\">12<\/td>\r\n<td style=\"width: 20%; height: 15px;\">12<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">0<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-1]^2\">0<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 10%;\">Wednesday<\/td>\r\n<td style=\"width: 10%; height: 15px;\">9<\/td>\r\n<td style=\"width: 20%; height: 15px;\">12<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-3<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 10%;\">Thursday<\/td>\r\n<td style=\"width: 10%; height: 15px;\">9<\/td>\r\n<td style=\"width: 20%; height: 15px;\">12<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-3<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\r\n<\/tr>\r\n<tr style=\"height: 31px;\">\r\n<td style=\"width: 10%;\">Friday<\/td>\r\n<td style=\"width: 10%; height: 31px;\">15<\/td>\r\n<td style=\"width: 20%; height: 31px;\">12<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">3<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow add the last column. $\\sum \\frac{(O-E)^2}{E} = 3$.\r\n\r\nTo find the <em data-effect=\"italics\">p<\/em>-value, calculate <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> &gt; 3). First notice that <em>df<\/em> = <em>k<\/em> - 1 = 5 - 1 = 4. This test is right-tailed. You can use the Google Sheets formula\r\n<code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\" default-formula-text-color\" dir=\"auto\">=<\/span><span class=\" default-formula-text-color\" dir=\"auto\">CHISQ.DIST.RT<\/span><span class=\" default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">3<\/span><span class=\" default-formula-text-color\" dir=\"auto\">,<\/span><span class=\"number\" dir=\"auto\">4<\/span><span class=\" default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code>\r\nto find <em>p<\/em>-value is approximately 0.5578\r\n<p id=\"element-894\">The <em data-effect=\"italics\">dfs<\/em> are the number of cells \u2013 1 = 5 \u2013 1 = 4<\/p>\r\n\r\n<div id=\"eip-idm102621328\" class=\"os-figure\">\r\n<figure data-id=\"eip-idm102621328\"><span id=\"id6207852\" data-type=\"media\" data-alt=\"This is a blank nonsymmetrical chi-square curve for the test statistic of the days of the week absent.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-559\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-300x141.jpg\" alt=\"This is a blank nonsymmetrical chi-square curve for the test statistic of the days of the week absent.\" width=\"300\" height=\"141\" \/><\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><span class=\"os-number\">10.4<\/span><\/div>\r\n<\/div>\r\n<p id=\"element-545\">The decision is not to reject the null hypothesis.<\/p>\r\n<p id=\"element-1000\"><strong>Conclusion:<\/strong> At a 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the absent days do not occur with equal frequencies.<\/p>\r\n\r\n<div class=\"textbox\">\r\n\r\nYou can use a <a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi-square distribution table<\/a> to find the range that contains the correct <em>p<\/em>-value. Usually, this is enough to draw a conclusion about a hypothesis test.\r\n\r\n<strong><em>p<\/em>-value Method<\/strong>\r\n\r\nIn this example, once you've calculated the test statistic, look in row 4 (since our <em>df<\/em> = 4). In this row, find the two numbers that are above and below our test statistic 3. Then follow those columns up to find which two values are above and below our <em>p<\/em>-value.\r\n\r\n<a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi2Distribution<\/a> <img class=\"aligncenter wp-image-670 size-full\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/chisq-pvalue.png\" alt=\"Snapshot of Chi Square Distribution table with row 4 circled, highlighting the values 1.064 and 7.779 leading up to the p-value lower range 0.1 and upper range 0.9\" width=\"971\" height=\"197\" \/>\r\n\r\nThe range of the <em>p<\/em>-value is 0.1 &lt; <em>p<\/em>-value &lt; 0.9 which tells us that it is larger than the significance level 0.05, so we fail to reject the null hypothesis (as we determined using Google Sheets as well).\r\n\r\n<strong>Critical Value Method<\/strong>\r\n\r\nAlternatively, we could get the critical value and compare it to the test statistic.\r\n\r\nFind the column with the significance level 0.05 in the top row, then go down to row 4 (<em>df<\/em> = 4) and that is your critical value.\r\n\r\n<img class=\"aligncenter size-full wp-image-671\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/chisq-critval.png\" alt=\"Snapshot of chi square distribtion table with significance level 0.05 circled and a line drawn down to row 4 containing the critical value 9.488\" width=\"964\" height=\"197\" \/>\r\n\r\nOur test statistic $\\chi^2 = 3$ is not larger than the critical value 9.488, so we fail to reject the null hypothesis.\r\n\r\n<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idm3127168\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.2<\/span><\/h3>\r\n<\/header><section>\r\n<div id=\"eip-723\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-375\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"eip-426\">Teachers want to know which night each week their students are doing most of their homework. Most teachers think that students do homework equally throughout the week. Suppose a random sample of 56 students were asked on which night of the week they did the most homework. The results were distributed as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl008\">Table 10.8<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl008\" class=\"os-table \">\r\n<table summary=\"Table 10.8 \" data-id=\"M03_Ch03_tbl008\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\"><\/th>\r\n<th scope=\"col\">Sunday<\/th>\r\n<th scope=\"col\">Monday<\/th>\r\n<th scope=\"col\">Tuesday<\/th>\r\n<th scope=\"col\">Wednesday<\/th>\r\n<th scope=\"col\">Thursday<\/th>\r\n<th scope=\"col\">Friday<\/th>\r\n<th scope=\"col\">Saturday<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Number of Students<\/td>\r\n<td>11<\/td>\r\n<td>8<\/td>\r\n<td>10<\/td>\r\n<td>7<\/td>\r\n<td>10<\/td>\r\n<td>5<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.8<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-idm100475616\">From the population of students, do the nights for the highest number of students doing the majority of their homework occur with equal frequencies during a week? What type of hypothesis test should you use?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div id=\"example3\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.3<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<p id=\"element-59\">One study indicates that the number of televisions that American families have is distributed (this is the <strong>given<\/strong> distribution for the American population) as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl009\">Table 10.9<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl009\" class=\"os-table \">\r\n<table summary=\"Table 10.9 \" data-id=\"M03_Ch03_tbl009\">\r\n<thead valign=\"top\">\r\n<tr>\r\n<th scope=\"col\">Number of Televisions<\/th>\r\n<th scope=\"col\">Percent<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>55<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4+<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.9<\/span><\/div>\r\n<\/div>\r\n<p id=\"element-23563\">The table contains expected (<em data-effect=\"italics\">E<\/em>) percents.<\/p>\r\n<p id=\"element-236\">A random sample of 600 families in the far western United States resulted in the data in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl010\">Table 10.10<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl010\" class=\"os-table \">\r\n<table summary=\"Table 10.10 \" data-id=\"M03_Ch03_tbl010\">\r\n<thead valign=\"top\">\r\n<tr>\r\n<th scope=\"col\">Number of Televisions<\/th>\r\n<th scope=\"col\">Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tfoot>\r\n<tr>\r\n<td><\/td>\r\n<td>Total = 600<\/td>\r\n<\/tr>\r\n<\/tfoot>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>66<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>119<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>340<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>60<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4+<\/td>\r\n<td>15<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.10<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-idm95551856\">The table contains observed (<em data-effect=\"italics\">O<\/em>) frequency values.<\/p>\r\n\r\n<div id=\"element-911\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id6229414\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-649\">At the 1% significance level, does it appear that the distribution \"number of televisions\" of far western United States families is different from the distribution for the American population as a whole?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"id6229434\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\r\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.3<\/span><\/h4>\r\n<div class=\"os-solution-container\">\r\n<p id=\"element-538\">This problem asks you to test whether the far western United States families distribution fits the distribution of the American families. This test is always right-tailed.<\/p>\r\n<p id=\"element-788\">The first table contains expected percentages. To get expected (<em data-effect=\"italics\">E<\/em>) frequencies, multiply the percentage by 600. The expected frequencies are shown in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl011\">Table 10.11<\/a>.<\/p>\r\n\r\n<div id=\"M03_Ch03_tbl011\" class=\"os-table \">\r\n<table summary=\"Table 10.11 \" data-id=\"M03_Ch03_tbl011\">\r\n<thead valign=\"top\">\r\n<tr>\r\n<th scope=\"col\">Number of Televisions<\/th>\r\n<th scope=\"col\">Percent<\/th>\r\n<th scope=\"col\">Expected Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>10<\/td>\r\n<td>(0.10)(600) = 60<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>16<\/td>\r\n<td>(0.16)(600) = 96<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>55<\/td>\r\n<td>(0.55)(600) = 330<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>11<\/td>\r\n<td>(0.11)(600) = 66<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>over 3<\/td>\r\n<td>8<\/td>\r\n<td>(0.08)(600) = 48<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.11<\/span><\/div>\r\n<\/div>\r\n<p id=\"element-whatever\" class=\"finger\">Therefore, the expected frequencies are 60, 96, 330, 66, and 48.<\/p>\r\n<p id=\"element-239\"><em data-effect=\"italics\">H<sub>0<\/sub><\/em>: The \"number of televisions\" distribution of far western United States families is the same as the \"number of televisions\" distribution of the American population.<\/p>\r\n<p id=\"element-2\"><em data-effect=\"italics\">H<sub>1<\/sub><\/em>: The \"number of televisions\" distribution of far western United States families is different from the \"number of televisions\" distribution of the American population.<\/p>\r\n<p id=\"element-228\">Distribution for the test: <span class=\"os-math-in-para\"><span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" style=\"position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;&amp;#x3C7;&lt;\/mi&gt;&lt;mn&gt;4&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/msubsup&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;\u03c7&lt;\/mi&gt;&lt;mn&gt;4&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/msubsup&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MathJax-Span-52\" class=\"math\" style=\"width: 1.387em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.052em; font-size: 129%;\"><span style=\"position: absolute; clip: rect(1.233em, 1001.05em, 2.695em, -1000em); top: -2.215em; left: 0em;\"><span id=\"MathJax-Span-53\" class=\"mrow\"><span id=\"MathJax-Span-54\" class=\"semantics\"><span id=\"MathJax-Span-55\" class=\"mrow\"><span id=\"MathJax-Span-56\" class=\"mrow\"><span id=\"MathJax-Span-57\" class=\"msubsup\"><span style=\"display: inline-block; position: relative; width: 1.055em;\"><span style=\"position: absolute; clip: rect(3.379em, 1000.6em, 4.357em, -1000em); top: -3.987em; left: 0em;\"><span id=\"MathJax-Span-58\" class=\"mi\" style=\"font-family: MathJax_Math; font-style: italic;\">\u03c7<\/span><\/span><span style=\"position: absolute; clip: rect(3.35em, 1000.43em, 4.153em, -1000em); top: -4.331em; left: 0.626em;\"><span id=\"MathJax-Span-59\" class=\"mn\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"position: absolute; clip: rect(3.342em, 1000.43em, 4.153em, -1000em); top: -3.672em; left: 0.626em;\"><span id=\"MathJax-Span-60\" class=\"mn\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\r\n$\\chi_4^2$ where <em data-effect=\"italics\">df<\/em> = (the number of cells) \u2013 1 = 5 \u2013 1 = 4.\r\n<div id=\"id6229977\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\" data-type=\"title\"><span id=\"21\" class=\"os-title-label\" data-type=\"\">Note<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-idm1659296\"><em data-effect=\"italics\">df<\/em> \u2260 600 \u2013 1<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"element-65\"><strong>Calculate the test statistic:<\/strong><\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 20%;\" scope=\"col\">Number of Televisions<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$O$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$E$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$O - E$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$(O - E)^2$<\/th>\r\n<th style=\"width: 16%; height: 31px;\">$\\frac{(O - E)^2}{E}$<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0494%;\">0<\/td>\r\n<td style=\"width: 10%;\">66<\/td>\r\n<td style=\"width: 20%; height: 15px;\">60<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:6}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">6<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:36}\" data-sheets-formula=\"=R[0]C[-1]^2\">36<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.6}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0494%;\">1<\/td>\r\n<td style=\"width: 10%;\">119<\/td>\r\n<td style=\"width: 20%; height: 15px;\">96<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:23}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">23<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:529}\" data-sheets-formula=\"=R[0]C[-1]^2\">529<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:5.510416666666667}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">5.510<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0494%;\">2<\/td>\r\n<td style=\"width: 10%;\">340<\/td>\r\n<td style=\"width: 20%; height: 15px;\">330<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:10}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">10<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:100}\" data-sheets-formula=\"=R[0]C[-1]^2\">100<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.30303030303030304}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.303<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0494%;\">3<\/td>\r\n<td style=\"width: 10%;\">60<\/td>\r\n<td style=\"width: 20%; height: 15px;\">66<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-6}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-6<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:36}\" data-sheets-formula=\"=R[0]C[-1]^2\">36<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.5454545454545454}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.545<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.0494%;\">over 3<\/td>\r\n<td style=\"width: 10%;\">15<\/td>\r\n<td style=\"width: 20%; height: 31px;\">48<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-33}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-33<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1089}\" data-sheets-formula=\"=R[0]C[-1]^2\">1089<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:22.6875}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">22.688<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow add the last column. $\\sum \\frac{(O-E)^2}{E} = 29.65$.\r\n<p id=\"element-635\"><strong>Find the critical value:<\/strong> Either use the <a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi-square distribution table<\/a> as described in the note at the bottom of Example 10.2 or use the following Google Sheets function noting the given significance level is 0.01 and the degrees of freedom <em>df<\/em> = <em>k<\/em> - 1 = 5 - 1 = 4:<em data-effect=\"italics\">\r\n<\/em><\/p>\r\n<code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\" default-formula-text-color\" dir=\"auto\">=<\/span><span class=\" default-formula-text-color\" dir=\"auto\">CHISQ.INV.RT<\/span><span class=\" default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">0.01<\/span><span class=\" default-formula-text-color\" dir=\"auto\">,<\/span><span class=\"number\" dir=\"auto\">4<\/span><span class=\" default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code>\r\n\r\nWe get a critical value 13.276\r\n<p id=\"element-42\"><strong>Compare the test statistic and critical value:<\/strong><\/p>\r\n\r\n<ul id=\"list-whatever3\">\r\n \t<li>test statistic $\\chi^2 = 29.65$<em data-effect=\"italics\">\r\n<\/em><\/li>\r\n \t<li>critical value is 13.276<em data-effect=\"italics\">\r\n<\/em><\/li>\r\n<\/ul>\r\n<p id=\"23\">So, the test statistic is further into the right tail than the critical value.<em data-effect=\"italics\">\r\n<\/em><\/p>\r\n<p id=\"element-251\"><strong>Make a decision:<\/strong> Since the test statistic is larger than (further to the right) the critical value, reject <em data-effect=\"italics\">H<sub>o<\/sub><\/em>.<\/p>\r\n<p id=\"element-717\">This means you reject the belief that the distribution for the far western states is the same as that of the American population as a whole.<\/p>\r\n<p id=\"element-387\"><strong>Conclusion:<\/strong> At the 1% significance level, from the data, there is sufficient evidence to conclude that the \"number of televisions\" distribution for the far western United States is different from the \"number of televisions\" distribution for the American population as a whole.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idm99666800\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.3<\/span><\/h3>\r\n<\/header><section>\r\n<div id=\"eip-521\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-211\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"eip-938\">The expected percentage of the number of pets students have in their homes is distributed (this is the given distribution for the student population of the United States) as in <a class=\"autogenerated-content\" href=\"#fs-idm147763264\">Table 10.12<\/a>.<\/p>\r\n\r\n<div id=\"fs-idm147763264\" class=\"os-table \">\r\n<table summary=\"Table 10.12 \" data-id=\"fs-idm147763264\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number of Pets<\/th>\r\n<th scope=\"col\">Percent<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>18<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>25<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>30<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>18<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4+<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.12<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-idp91504704\">A random sample of 1,000 students from the Eastern United States resulted in the data in <a class=\"autogenerated-content\" href=\"#fs-idm64242624\">Table 10.13<\/a>.<\/p>\r\n\r\n<div id=\"fs-idm64242624\" class=\"os-table \">\r\n<table summary=\"Table 10.13 \" data-id=\"fs-idm64242624\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Number of Pets<\/th>\r\n<th scope=\"col\">Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>210<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>240<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>320<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>140<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4+<\/td>\r\n<td>90<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.13<\/span><\/div>\r\n<\/div>\r\n<p id=\"eip-idp27106912\">At the 1% significance level, does it appear that the distribution \u201cnumber of pets\u201d of students in the Eastern United States is different from the distribution for the United States student population as a whole? What is the <em data-effect=\"italics\">p<\/em>-value?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div id=\"element-866\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.4<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"element-221\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id6230511\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-95\">Suppose you flip two coins 100 times. The results are 20 <em data-effect=\"italics\">HH<\/em>, 27 <em data-effect=\"italics\">HT<\/em>, 30 <em data-effect=\"italics\">TH<\/em>, and 23 <em data-effect=\"italics\">TT<\/em>. Are the coins fair? Test at a 5% significance level.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"id6230530\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\r\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.4<\/span><\/h4>\r\n<div class=\"os-solution-container\">\r\n<p id=\"element-929\">This problem can be set up as a goodness-of-fit problem. The sample space for flipping two fair coins is {<em data-effect=\"italics\">HH<\/em>, <em data-effect=\"italics\">HT<\/em>, <em data-effect=\"italics\">TH<\/em>, <em data-effect=\"italics\">TT<\/em>}. Out of 100 flips, you would expect 25 <em data-effect=\"italics\">HH<\/em>, 25 <em data-effect=\"italics\">HT<\/em>, 25 <em data-effect=\"italics\">TH<\/em>, and 25 <em data-effect=\"italics\">TT<\/em>. This is the expected distribution. The question, \"Are the coins fair?\" is the same as saying, \"Does the distribution of the coins (20 <em data-effect=\"italics\">HH<\/em>, 27 <em data-effect=\"italics\">HT<\/em>, 30 <em data-effect=\"italics\">TH<\/em>, 23 <em data-effect=\"italics\">TT<\/em>) fit the expected distribution?\"<\/p>\r\n<p id=\"element-570\"><strong>Random Variable:<\/strong> Let <em data-effect=\"italics\">X<\/em> = the number of heads in one flip of the two coins. <em data-effect=\"italics\">X<\/em> takes on the values 0, 1, 2. (There are 0, 1, or 2 heads in the flip of two coins.) Therefore, the <strong>number of cells is three<\/strong>. Since <em data-effect=\"italics\">X<\/em> = the number of heads, the observed frequencies are 20 (for two heads, HH), 57 (for one head, which is the HT and TH counts added together), and 23 (for zero heads or both tails, TT). The expected frequencies are 25 (for two heads), 50 (for one head), and 25 (for zero heads or both tails). This test is right-tailed.<\/p>\r\n<p id=\"element-774\"><strong><em data-effect=\"italics\">H<sub>0<\/sub><\/em><\/strong>: The coins are fair.<\/p>\r\n<p id=\"element-734\"><strong><em data-effect=\"italics\">H<sub>1<\/sub><\/em><\/strong>: The coins are not fair.<\/p>\r\n<p id=\"element-136\"><strong>Distribution for the test:<\/strong><\/p>\r\n$\\chi_2^2$ where <em data-effect=\"italics\">df<\/em> = 3 \u2013 1 = 2.\r\n<p id=\"element-380\"><strong>Calculate the test statistic:\r\n<\/strong><\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 31px;\">\r\n<th style=\"width: 10%;\"># of heads<\/th>\r\n<th style=\"width: 10%; height: 31px;\">$O$<\/th>\r\n<th style=\"width: 20%; height: 31px;\">$E$<\/th>\r\n<th style=\"width: 20%; height: 31px;\">$O - E$<\/th>\r\n<th style=\"width: 20%; height: 31px;\">$(O - E)^2$<\/th>\r\n<th style=\"width: 20%; height: 31px;\">$\\frac{(O - E)^2}{E}$<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10%;\">0<\/td>\r\n<td style=\"width: 10%;\">20<\/td>\r\n<td style=\"width: 20%; height: 15px;\">25<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-5}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-5<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:25}\" data-sheets-formula=\"=R[0]C[-1]^2\">25<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10%;\">1<\/td>\r\n<td style=\"width: 10%;\">57<\/td>\r\n<td style=\"width: 20%; height: 15px;\">50<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:2}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">7<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:4}\" data-sheets-formula=\"=R[0]C[-1]^2\">49<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.16}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.98<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10%;\">2<\/td>\r\n<td style=\"width: 10%;\">23<\/td>\r\n<td style=\"width: 20%; height: 15px;\">25<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:5}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-2<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:25}\" data-sheets-formula=\"=R[0]C[-1]^2\">4<\/td>\r\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> = 2.14\r\n<p id=\"element-490\"><strong>Probability statement:<\/strong> <em data-effect=\"italics\">p<\/em>-value = <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> &gt; 2.14) = 0.343 using the Google Sheets function\r\n<code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\" default-formula-text-color\" dir=\"auto\">=<\/span><span class=\" default-formula-text-color\" dir=\"auto\">CHISQ.DIST.RT<\/span><span class=\" default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">2.14<\/span><span class=\" default-formula-text-color\" dir=\"auto\">,<\/span>2<span class=\" default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code><\/p>\r\n<p id=\"element-817\"><strong>Compare <em data-effect=\"italics\">\u03b1<\/em> and the <em data-effect=\"italics\">p<\/em>-value:<\/strong><\/p>\r\n\r\n<ul id=\"element-xyz\">\r\n \t<li><em data-effect=\"italics\">\u03b1<\/em> = 0.05<\/li>\r\n \t<li><em data-effect=\"italics\">p<\/em>-value = 0.343<\/li>\r\n<\/ul>\r\n<p id=\"64\"><em data-effect=\"italics\">\u03b1<\/em> &lt; <em data-effect=\"italics\">p<\/em>-value.<\/p>\r\n<p id=\"element-58\"><strong>Make a decision:<\/strong> Since <em data-effect=\"italics\">\u03b1<\/em> &lt; <em data-effect=\"italics\">p<\/em>-value, do not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>.<\/p>\r\n<p id=\"element-995\"><strong>Conclusion:<\/strong> There is insufficient evidence to conclude that the coins are not fair.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idm153122384\" class=\"statistics try finger ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.4<\/span><\/h3>\r\n<\/header><section>\r\n<div id=\"eip-860\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-476\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"eip-907\">Students in a social studies class hypothesize that the literacy rates across the world for every region are 82%. <a class=\"autogenerated-content\" href=\"#fs-idm84960960\">Table 10.14<\/a> shows the actual literacy rates across the world broken down by region. What are the test statistic and the degrees of freedom?<\/p>\r\n\r\n<div id=\"fs-idm84960960\" class=\"os-table \">\r\n<table summary=\"Table 10.14 \" data-id=\"fs-idm84960960\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">MDG Region<\/th>\r\n<th scope=\"col\">Adult Literacy Rate (%)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Developed Regions<\/td>\r\n<td>99.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Commonwealth of Independent States<\/td>\r\n<td>99.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Northern Africa<\/td>\r\n<td>67.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sub-Saharan Africa<\/td>\r\n<td>62.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Latin America and the Caribbean<\/td>\r\n<td>91.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Eastern Asia<\/td>\r\n<td>93.8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Southern Asia<\/td>\r\n<td>61.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>South-Eastern Asia<\/td>\r\n<td>91.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Western Asia<\/td>\r\n<td>84.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Oceania<\/td>\r\n<td>66.4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.14<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<p><span style=\"display: none;\"><br \/>\n[latexpage]<br \/>\n<\/span><\/p>\n<div id=\"d2406e86-3ba5-463c-a86b-7e7e02a03a4e\" class=\"chapter-content-module\" data-type=\"page\" data-cnxml-to-html-ver=\"2.1.0\">\n<p id=\"element-112\">In this type of hypothesis test, you determine whether the data <strong>&#8220;fit&#8221;<\/strong> a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. <strong>The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.<\/strong><\/p>\n<p id=\"element-908\">The test statistic for a goodness-of-fit test is:<\/p>\n<p>$$\\sum_k \\frac{(O-E)^2}{E}$$<\/p>\n<p>where:<\/p>\n<ul id=\"fs-idm37230752\">\n<li><em data-effect=\"italics\">O<\/em> = <span id=\"term206\" data-type=\"term\">observed values<\/span> (data)<\/li>\n<li id=\"element-481\"><em data-effect=\"italics\">E<\/em> = <span id=\"term207\" data-type=\"term\">expected values<\/span> (from theory)<\/li>\n<li id=\"element-402\"><em data-effect=\"italics\">k<\/em> = the number of different data cells or categories<\/li>\n<\/ul>\n<p id=\"element-512\"><strong>The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true.<\/strong> There are <em data-effect=\"italics\">k<\/em> terms of the form $\\frac{(O-E)^2}{E}$.<\/p>\n<p>The number of degrees of freedom is <em data-effect=\"italics\">df<\/em> = (number of categories \u2013 1) = <em>k<\/em> &#8211; 1<\/p>\n<p id=\"element-838\"><strong>The <span id=\"term208\" data-type=\"term\">goodness-of-fit test<\/span> is almost always right-tailed.<\/strong> If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.<\/p>\n<div id=\"eip-587\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\" data-type=\"title\"><span id=\"1\" class=\"os-title-label\" data-type=\"\">Note<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-idm98728224\">The expected value for each cell needs to be at least five in order for you to use this test.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"element-719\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.1<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"body\">\n<p id=\"element-615\">Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism rate follows faculty perception. The faculty expected that a group of 100 students would miss class according to <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl001\">Table 10.1<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl001\" class=\"os-table\">\n<table summary=\"Table 10.1\" data-id=\"M03_Ch03_tbl001\">\n<thead>\n<tr>\n<th scope=\"col\">Number of absences per term<\/th>\n<th scope=\"col\">Expected number of students<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u20132<\/td>\n<td>50<\/td>\n<\/tr>\n<tr>\n<td>3\u20135<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td>6\u20138<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>9\u201311<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>12+<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.1<\/span><\/div>\n<\/div>\n<p id=\"eip-952\">A random survey across all mathematics courses was then done to determine the actual number <strong>(observed)<\/strong> of absences in a course. The chart in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl002\">Table 10.2<\/a> displays the results of that survey.<\/p>\n<div id=\"M03_Ch03_tbl002\" class=\"os-table\">\n<table summary=\"Table 10.2\" data-id=\"M03_Ch03_tbl002\">\n<thead>\n<tr>\n<th scope=\"col\">Number of absences per term<\/th>\n<th scope=\"col\">Actual number of students<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u20132<\/td>\n<td>35<\/td>\n<\/tr>\n<tr>\n<td>3\u20135<\/td>\n<td>40<\/td>\n<\/tr>\n<tr>\n<td>6\u20138<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>9\u201311<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>12+<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.2<\/span><\/div>\n<\/div>\n<p id=\"2\"><span data-type=\"newline\"><br \/>\n<\/span>Determine the null and alternative hypotheses needed to conduct a goodness-of-fit test.<\/p>\n<p id=\"element-846\"><strong><em data-effect=\"italics\">H<sub>0<\/sub><\/em>:<\/strong> Student absenteeism <strong>fits<\/strong> faculty perception.<\/p>\n<p id=\"element-82\"><span data-type=\"newline\"><br \/>\n<\/span>The alternative hypothesis is the opposite of the null hypothesis.<\/p>\n<p id=\"element-421\"><strong><em data-effect=\"italics\">H<sub>1<\/sub><\/em>:<\/strong> Student absenteeism <strong>does not fit<\/strong> faculty perception.<\/p>\n<div id=\"fs-idp16583968\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id6198828\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-451\">a. Can you use the information as it appears in the charts to conduct the goodness-of-fit test?<\/p>\n<\/div>\n<\/div>\n<div id=\"id6198852\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.1<\/span><\/h4>\n<div class=\"os-solution-container\">\n<p id=\"element-600\">a. <strong>No.<\/strong> Notice that the expected number of absences for the &#8220;12+&#8221; entry is less than five (it is two). Combine that group with the &#8220;9\u201311&#8221; group to create new tables where the number of students for each entry are at least five. The new results are in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl003\">Table 10.3<\/a> and <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl004\">Table 10.4<\/a>.<br \/>\n<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"M03_Ch03_tbl003\" class=\"os-table\">\n<table summary=\"Table 10.3\" data-id=\"M03_Ch03_tbl003\">\n<thead>\n<tr>\n<th scope=\"col\"><strong>Number of absences per term<\/strong><\/th>\n<th scope=\"col\"><strong>Expected number of students<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u20132<\/td>\n<td>50<\/td>\n<\/tr>\n<tr>\n<td>3\u20135<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td>6\u20138<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>9+<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.3<\/span><\/div>\n<\/div>\n<div id=\"M03_Ch03_tbl004\" class=\"os-table\">\n<table summary=\"Table 10.4\" data-id=\"M03_Ch03_tbl004\">\n<thead>\n<tr>\n<th scope=\"col\">Number of absences per term<\/th>\n<th scope=\"col\">Actual number of students<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u20132<\/td>\n<td>35<\/td>\n<\/tr>\n<tr>\n<td>3\u20135<\/td>\n<td>40<\/td>\n<\/tr>\n<tr>\n<td>6\u20138<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>9+<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.4<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"element-612\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id6198893\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-496\">b. What is the number of degrees of freedom (<em data-effect=\"italics\">df<\/em>)?<\/p>\n<\/div>\n<\/div>\n<div id=\"id6198915\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.1<\/span><\/h4>\n<div class=\"os-solution-container\">\n<p id=\"element-601\">b. There are four &#8220;cells&#8221; or categories in each of the new tables.<\/p>\n<p id=\"element-6012\"><em data-effect=\"italics\">df<\/em> = number of cells \u2013 1 = 4 \u2013 1 = 3<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idm289894560\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.1<\/span><\/h3>\n<\/header>\n<section>\n<div id=\"eip-31\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"eip-864\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"eip-988\">A factory manager needs to understand how many products are defective versus how many are produced. The number of expected defects is listed in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl005\">Table 10.5<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl005\" class=\"os-table\">\n<table summary=\"Table 10.5\" data-id=\"M03_Ch03_tbl005\">\n<thead>\n<tr>\n<th scope=\"col\">Number produced<\/th>\n<th scope=\"col\">Number defective<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u2013100<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>101\u2013200<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>201\u2013300<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>301\u2013400<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>401\u2013500<\/td>\n<td>10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.5<\/span><\/div>\n<\/div>\n<p id=\"eip-idm111370528\">A random sample was taken to determine the actual number of defects. <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl006\">Table 10.6<\/a> shows the results of the survey.<\/p>\n<div id=\"M03_Ch03_tbl006\" class=\"os-table\">\n<table summary=\"Table 10.6\" data-id=\"M03_Ch03_tbl006\">\n<thead>\n<tr>\n<th scope=\"col\">Number produced<\/th>\n<th scope=\"col\">Number defective<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u2013100<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>101\u2013200<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>201\u2013300<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>301\u2013400<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>401\u2013500<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.6<\/span><\/div>\n<\/div>\n<p id=\"eip-idp74195536\">State the null and alternative hypotheses needed to conduct a goodness-of-fit test, and state the degrees of freedom.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<p>In the following Example 10.2, our solution will use the <strong><em>p<\/em>-value method<\/strong> where we find the <em>p<\/em>-value corresponding to our test statistic and compare it with the significance level given.<\/p>\n<p>In Example 10.3, our solution will use the <strong>critical value method<\/strong> where we find the critical value corresponding to the given significance level, and compare the critical value to the test statistic.<\/p>\n<p>The <em>p<\/em>-value method and the critical value method are equivalent when determining whether to reject or fail to reject the null hypothesis.<\/p>\n<div id=\"element-962\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.2<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"element-811\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id6215628\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-733\">Employers want to know which days of the week employees are absent in a five-day work week. Most employers would like to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week they had the highest number of employee absences. The results were distributed as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl007\">Table 10.7<\/a>. For the population of employees, do the days for the highest number of absences occur with equal frequencies during a five-day work week? Test at a 5% significance level.<\/p>\n<div id=\"M03_Ch03_tbl007\" class=\"os-table\">\n<table summary=\"Table 10.7 Day of the Week Employees were Most Absent\" data-id=\"M03_Ch03_tbl007\">\n<thead valign=\"top\">\n<tr>\n<th scope=\"col\"><\/th>\n<th scope=\"col\">Monday<\/th>\n<th scope=\"col\">Tuesday<\/th>\n<th scope=\"col\">Wednesday<\/th>\n<th scope=\"col\">Thursday<\/th>\n<th scope=\"col\">Friday<\/th>\n<\/tr>\n<\/thead>\n<tbody valign=\"top\">\n<tr>\n<td>Number of Absences<\/td>\n<td>15<\/td>\n<td>12<\/td>\n<td>9<\/td>\n<td>9<\/td>\n<td>15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.7<\/span> <span class=\"os-title\" data-type=\"title\">Day of the Week Employees were Most Absent<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"id6215647\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.2<\/span><\/h4>\n<div class=\"os-solution-container\">\n<p id=\"element-13\">The null and alternative hypotheses are:<\/p>\n<ul id=\"element-741\">\n<li><em data-effect=\"italics\">H<sub>0<\/sub><\/em>: The absent days occur with equal frequencies, that is, they fit a uniform distribution.<\/li>\n<li><em data-effect=\"italics\">H<sub>1<\/sub><\/em>: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution.<\/li>\n<\/ul>\n<p id=\"element-71\">If the absent days occur with equal frequencies, then, out of 60 absent days (the total in the sample: 15 + 12 + 9 + 9 + 15 = 60), there would be 12 absences on Monday, 12 on Tuesday, 12 on Wednesday, 12 on Thursday, and 12 on Friday. These numbers are the <strong>expected<\/strong> (<em data-effect=\"italics\">E<\/em>) values. The values in the table are the <strong>observed<\/strong> (<em data-effect=\"italics\">O<\/em>) values or data.<\/p>\n<p id=\"element-489\">This time, calculate the <em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> test statistic by hand. Make a chart with the following headings and fill in the columns:<\/p>\n<ul id=\"element-120\">\n<li>Expected (<em data-effect=\"italics\">E<\/em>) values (12, 12, 12, 12, 12)<\/li>\n<li>Observed (<em data-effect=\"italics\">O<\/em>) values (15, 12, 9, 9, 15)<\/li>\n<li>(<em data-effect=\"italics\">O<\/em> \u2013 <em data-effect=\"italics\">E<\/em>)<\/li>\n<li>(<em data-effect=\"italics\">O<\/em> \u2013 <em data-effect=\"italics\">E<\/em>)<sup>2<\/sup><\/li>\n<li>$\\frac{(O-E)^2}{E}$<\/li>\n<\/ul>\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\">\n<tbody>\n<tr style=\"height: 31px;\">\n<th style=\"width: 20%;\">Day of Absences<\/th>\n<th style=\"width: 16%; height: 31px;\">$O$<\/th>\n<th style=\"width: 16%; height: 31px;\">$E$<\/th>\n<th style=\"width: 16%; height: 31px;\">$O &#8211; E$<\/th>\n<th style=\"width: 16%; height: 31px;\">$(O &#8211; E)^2$<\/th>\n<th style=\"width: 16%; height: 31px;\">$\\frac{(O &#8211; E)^2}{E}$<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 10%;\">Monday<\/td>\n<td style=\"width: 10%; height: 15px;\">15<\/td>\n<td style=\"width: 20%; height: 15px;\">12<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">3<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 10%;\">Tuesday<\/td>\n<td style=\"width: 10%; height: 15px;\">12<\/td>\n<td style=\"width: 20%; height: 15px;\">12<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">0<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-1]^2\">0<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 10%;\">Wednesday<\/td>\n<td style=\"width: 10%; height: 15px;\">9<\/td>\n<td style=\"width: 20%; height: 15px;\">12<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-3<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 10%;\">Thursday<\/td>\n<td style=\"width: 10%; height: 15px;\">9<\/td>\n<td style=\"width: 20%; height: 15px;\">12<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-3<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\n<\/tr>\n<tr style=\"height: 31px;\">\n<td style=\"width: 10%;\">Friday<\/td>\n<td style=\"width: 10%; height: 31px;\">15<\/td>\n<td style=\"width: 20%; height: 31px;\">12<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:3}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">3<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:9}\" data-sheets-formula=\"=R[0]C[-1]^2\">9<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.75}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.75<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now add the last column. $\\sum \\frac{(O-E)^2}{E} = 3$.<\/p>\n<p>To find the <em data-effect=\"italics\">p<\/em>-value, calculate <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> &gt; 3). First notice that <em>df<\/em> = <em>k<\/em> &#8211; 1 = 5 &#8211; 1 = 4. This test is right-tailed. You can use the Google Sheets formula<br \/>\n<code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\"default-formula-text-color\" dir=\"auto\">=<\/span><span class=\"default-formula-text-color\" dir=\"auto\">CHISQ.DIST.RT<\/span><span class=\"default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">3<\/span><span class=\"default-formula-text-color\" dir=\"auto\">,<\/span><span class=\"number\" dir=\"auto\">4<\/span><span class=\"default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code><br \/>\nto find <em>p<\/em>-value is approximately 0.5578<\/p>\n<p id=\"element-894\">The <em data-effect=\"italics\">dfs<\/em> are the number of cells \u2013 1 = 5 \u2013 1 = 4<\/p>\n<div id=\"eip-idm102621328\" class=\"os-figure\">\n<figure data-id=\"eip-idm102621328\"><span id=\"id6207852\" data-type=\"media\" data-alt=\"This is a blank nonsymmetrical chi-square curve for the test statistic of the days of the week absent.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-559\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-300x141.jpg\" alt=\"This is a blank nonsymmetrical chi-square curve for the test statistic of the days of the week absent.\" width=\"300\" height=\"141\" srcset=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-300x141.jpg 300w, https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-65x31.jpg 65w, https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-225x106.jpg 225w, https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d-350x165.jpg 350w, https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/039a6a3fbbc667b9b7ec97f2dbb5b5840ef6437d.jpg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><span class=\"os-number\">10.4<\/span><\/div>\n<\/div>\n<p id=\"element-545\">The decision is not to reject the null hypothesis.<\/p>\n<p id=\"element-1000\"><strong>Conclusion:<\/strong> At a 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the absent days do not occur with equal frequencies.<\/p>\n<div class=\"textbox\">\n<p>You can use a <a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi-square distribution table<\/a> to find the range that contains the correct <em>p<\/em>-value. Usually, this is enough to draw a conclusion about a hypothesis test.<\/p>\n<p><strong><em>p<\/em>-value Method<\/strong><\/p>\n<p>In this example, once you&#8217;ve calculated the test statistic, look in row 4 (since our <em>df<\/em> = 4). In this row, find the two numbers that are above and below our test statistic 3. Then follow those columns up to find which two values are above and below our <em>p<\/em>-value.<\/p>\n<p><a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi2Distribution<\/a> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-670 size-full\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/chisq-pvalue.png\" alt=\"Snapshot of Chi Square Distribution table with row 4 circled, highlighting the values 1.064 and 7.779 leading up to the p-value lower range 0.1 and upper range 0.9\" width=\"971\" height=\"197\" \/><\/p>\n<p>The range of the <em>p<\/em>-value is 0.1 &lt; <em>p<\/em>-value &lt; 0.9 which tells us that it is larger than the significance level 0.05, so we fail to reject the null hypothesis (as we determined using Google Sheets as well).<\/p>\n<p><strong>Critical Value Method<\/strong><\/p>\n<p>Alternatively, we could get the critical value and compare it to the test statistic.<\/p>\n<p>Find the column with the significance level 0.05 in the top row, then go down to row 4 (<em>df<\/em> = 4) and that is your critical value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-671\" src=\"https:\/\/textbooks.jaykesler.net\/introstats\/wp-content\/uploads\/sites\/2\/2021\/01\/chisq-critval.png\" alt=\"Snapshot of chi square distribtion table with significance level 0.05 circled and a line drawn down to row 4 containing the critical value 9.488\" width=\"964\" height=\"197\" \/><\/p>\n<p>Our test statistic $\\chi^2 = 3$ is not larger than the critical value 9.488, so we fail to reject the null hypothesis.<\/p>\n<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idm3127168\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.2<\/span><\/h3>\n<\/header>\n<section>\n<div id=\"eip-723\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-375\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"eip-426\">Teachers want to know which night each week their students are doing most of their homework. Most teachers think that students do homework equally throughout the week. Suppose a random sample of 56 students were asked on which night of the week they did the most homework. The results were distributed as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl008\">Table 10.8<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl008\" class=\"os-table\">\n<table summary=\"Table 10.8\" data-id=\"M03_Ch03_tbl008\">\n<thead>\n<tr>\n<th scope=\"col\"><\/th>\n<th scope=\"col\">Sunday<\/th>\n<th scope=\"col\">Monday<\/th>\n<th scope=\"col\">Tuesday<\/th>\n<th scope=\"col\">Wednesday<\/th>\n<th scope=\"col\">Thursday<\/th>\n<th scope=\"col\">Friday<\/th>\n<th scope=\"col\">Saturday<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Number of Students<\/td>\n<td>11<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<td>7<\/td>\n<td>10<\/td>\n<td>5<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.8<\/span><\/div>\n<\/div>\n<p id=\"eip-idm100475616\">From the population of students, do the nights for the highest number of students doing the majority of their homework occur with equal frequencies during a week? What type of hypothesis test should you use?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"example3\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.3<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"body\">\n<p id=\"element-59\">One study indicates that the number of televisions that American families have is distributed (this is the <strong>given<\/strong> distribution for the American population) as in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl009\">Table 10.9<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl009\" class=\"os-table\">\n<table summary=\"Table 10.9\" data-id=\"M03_Ch03_tbl009\">\n<thead valign=\"top\">\n<tr>\n<th scope=\"col\">Number of Televisions<\/th>\n<th scope=\"col\">Percent<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>55<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>4+<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.9<\/span><\/div>\n<\/div>\n<p id=\"element-23563\">The table contains expected (<em data-effect=\"italics\">E<\/em>) percents.<\/p>\n<p id=\"element-236\">A random sample of 600 families in the far western United States resulted in the data in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl010\">Table 10.10<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl010\" class=\"os-table\">\n<table summary=\"Table 10.10\" data-id=\"M03_Ch03_tbl010\">\n<thead valign=\"top\">\n<tr>\n<th scope=\"col\">Number of Televisions<\/th>\n<th scope=\"col\">Frequency<\/th>\n<\/tr>\n<\/thead>\n<tfoot>\n<tr>\n<td><\/td>\n<td>Total = 600<\/td>\n<\/tr>\n<\/tfoot>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>66<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>119<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>340<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>60<\/td>\n<\/tr>\n<tr>\n<td>4+<\/td>\n<td>15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.10<\/span><\/div>\n<\/div>\n<p id=\"fs-idm95551856\">The table contains observed (<em data-effect=\"italics\">O<\/em>) frequency values.<\/p>\n<div id=\"element-911\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id6229414\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-649\">At the 1% significance level, does it appear that the distribution &#8220;number of televisions&#8221; of far western United States families is different from the distribution for the American population as a whole?<\/p>\n<\/div>\n<\/div>\n<div id=\"id6229434\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.3<\/span><\/h4>\n<div class=\"os-solution-container\">\n<p id=\"element-538\">This problem asks you to test whether the far western United States families distribution fits the distribution of the American families. This test is always right-tailed.<\/p>\n<p id=\"element-788\">The first table contains expected percentages. To get expected (<em data-effect=\"italics\">E<\/em>) frequencies, multiply the percentage by 600. The expected frequencies are shown in <a class=\"autogenerated-content\" href=\"#M03_Ch03_tbl011\">Table 10.11<\/a>.<\/p>\n<div id=\"M03_Ch03_tbl011\" class=\"os-table\">\n<table summary=\"Table 10.11\" data-id=\"M03_Ch03_tbl011\">\n<thead valign=\"top\">\n<tr>\n<th scope=\"col\">Number of Televisions<\/th>\n<th scope=\"col\">Percent<\/th>\n<th scope=\"col\">Expected Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>10<\/td>\n<td>(0.10)(600) = 60<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>16<\/td>\n<td>(0.16)(600) = 96<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>55<\/td>\n<td>(0.55)(600) = 330<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>11<\/td>\n<td>(0.11)(600) = 66<\/td>\n<\/tr>\n<tr>\n<td>over 3<\/td>\n<td>8<\/td>\n<td>(0.08)(600) = 48<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.11<\/span><\/div>\n<\/div>\n<p id=\"element-whatever\" class=\"finger\">Therefore, the expected frequencies are 60, 96, 330, 66, and 48.<\/p>\n<p id=\"element-239\"><em data-effect=\"italics\">H<sub>0<\/sub><\/em>: The &#8220;number of televisions&#8221; distribution of far western United States families is the same as the &#8220;number of televisions&#8221; distribution of the American population.<\/p>\n<p id=\"element-2\"><em data-effect=\"italics\">H<sub>1<\/sub><\/em>: The &#8220;number of televisions&#8221; distribution of far western United States families is different from the &#8220;number of televisions&#8221; distribution of the American population.<\/p>\n<p id=\"element-228\">Distribution for the test: <span class=\"os-math-in-para\"><span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" style=\"position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;&amp;#x3C7;&lt;\/mi&gt;&lt;mn&gt;4&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/msubsup&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;\u03c7&lt;\/mi&gt;&lt;mn&gt;4&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/msubsup&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MathJax-Span-52\" class=\"math\" style=\"width: 1.387em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.052em; font-size: 129%;\"><span style=\"position: absolute; clip: rect(1.233em, 1001.05em, 2.695em, -1000em); top: -2.215em; left: 0em;\"><span id=\"MathJax-Span-53\" class=\"mrow\"><span id=\"MathJax-Span-54\" class=\"semantics\"><span id=\"MathJax-Span-55\" class=\"mrow\"><span id=\"MathJax-Span-56\" class=\"mrow\"><span id=\"MathJax-Span-57\" class=\"msubsup\"><span style=\"display: inline-block; position: relative; width: 1.055em;\"><span style=\"position: absolute; clip: rect(3.379em, 1000.6em, 4.357em, -1000em); top: -3.987em; left: 0em;\"><span id=\"MathJax-Span-58\" class=\"mi\" style=\"font-family: MathJax_Math; font-style: italic;\">\u03c7<\/span><\/span><span style=\"position: absolute; clip: rect(3.35em, 1000.43em, 4.153em, -1000em); top: -4.331em; left: 0.626em;\"><span id=\"MathJax-Span-59\" class=\"mn\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><\/span><span style=\"position: absolute; clip: rect(3.342em, 1000.43em, 4.153em, -1000em); top: -3.672em; left: 0.626em;\"><span id=\"MathJax-Span-60\" class=\"mn\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>$\\chi_4^2$ where <em data-effect=\"italics\">df<\/em> = (the number of cells) \u2013 1 = 5 \u2013 1 = 4.<\/p>\n<div id=\"id6229977\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\" data-type=\"title\"><span id=\"21\" class=\"os-title-label\" data-type=\"\">Note<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-idm1659296\"><em data-effect=\"italics\">df<\/em> \u2260 600 \u2013 1<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"element-65\"><strong>Calculate the test statistic:<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\">\n<tbody>\n<tr>\n<th style=\"width: 20%;\" scope=\"col\">Number of Televisions<\/th>\n<th style=\"width: 16%; height: 31px;\">$O$<\/th>\n<th style=\"width: 16%; height: 31px;\">$E$<\/th>\n<th style=\"width: 16%; height: 31px;\">$O &#8211; E$<\/th>\n<th style=\"width: 16%; height: 31px;\">$(O &#8211; E)^2$<\/th>\n<th style=\"width: 16%; height: 31px;\">$\\frac{(O &#8211; E)^2}{E}$<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 16.0494%;\">0<\/td>\n<td style=\"width: 10%;\">66<\/td>\n<td style=\"width: 20%; height: 15px;\">60<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:6}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">6<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:36}\" data-sheets-formula=\"=R[0]C[-1]^2\">36<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.6}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0494%;\">1<\/td>\n<td style=\"width: 10%;\">119<\/td>\n<td style=\"width: 20%; height: 15px;\">96<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:23}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">23<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:529}\" data-sheets-formula=\"=R[0]C[-1]^2\">529<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:5.510416666666667}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">5.510<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0494%;\">2<\/td>\n<td style=\"width: 10%;\">340<\/td>\n<td style=\"width: 20%; height: 15px;\">330<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:10}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">10<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:100}\" data-sheets-formula=\"=R[0]C[-1]^2\">100<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.30303030303030304}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.303<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0494%;\">3<\/td>\n<td style=\"width: 10%;\">60<\/td>\n<td style=\"width: 20%; height: 15px;\">66<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-6}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-6<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:36}\" data-sheets-formula=\"=R[0]C[-1]^2\">36<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.5454545454545454}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.545<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.0494%;\">over 3<\/td>\n<td style=\"width: 10%;\">15<\/td>\n<td style=\"width: 20%; height: 31px;\">48<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-33}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-33<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1089}\" data-sheets-formula=\"=R[0]C[-1]^2\">1089<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:22.6875}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">22.688<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now add the last column. $\\sum \\frac{(O-E)^2}{E} = 29.65$.<\/p>\n<p id=\"element-635\"><strong>Find the critical value:<\/strong> Either use the <a href=\"https:\/\/textbooks.jaykesler.net\/introstats\/back-matter\/chi2-distribution\/\" target=\"_blank\" rel=\"noopener\">Chi-square distribution table<\/a> as described in the note at the bottom of Example 10.2 or use the following Google Sheets function noting the given significance level is 0.01 and the degrees of freedom <em>df<\/em> = <em>k<\/em> &#8211; 1 = 5 &#8211; 1 = 4:<em data-effect=\"italics\"><br \/>\n<\/em><\/p>\n<p><code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\"default-formula-text-color\" dir=\"auto\">=<\/span><span class=\"default-formula-text-color\" dir=\"auto\">CHISQ.INV.RT<\/span><span class=\"default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">0.01<\/span><span class=\"default-formula-text-color\" dir=\"auto\">,<\/span><span class=\"number\" dir=\"auto\">4<\/span><span class=\"default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code><\/p>\n<p>We get a critical value 13.276<\/p>\n<p id=\"element-42\"><strong>Compare the test statistic and critical value:<\/strong><\/p>\n<ul id=\"list-whatever3\">\n<li>test statistic $\\chi^2 = 29.65$<em data-effect=\"italics\"><br \/>\n<\/em><\/li>\n<li>critical value is 13.276<em data-effect=\"italics\"><br \/>\n<\/em><\/li>\n<\/ul>\n<p id=\"23\">So, the test statistic is further into the right tail than the critical value.<em data-effect=\"italics\"><br \/>\n<\/em><\/p>\n<p id=\"element-251\"><strong>Make a decision:<\/strong> Since the test statistic is larger than (further to the right) the critical value, reject <em data-effect=\"italics\">H<sub>o<\/sub><\/em>.<\/p>\n<p id=\"element-717\">This means you reject the belief that the distribution for the far western states is the same as that of the American population as a whole.<\/p>\n<p id=\"element-387\"><strong>Conclusion:<\/strong> At the 1% significance level, from the data, there is sufficient evidence to conclude that the &#8220;number of televisions&#8221; distribution for the far western United States is different from the &#8220;number of televisions&#8221; distribution for the American population as a whole.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idm99666800\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.3<\/span><\/h3>\n<\/header>\n<section>\n<div id=\"eip-521\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-211\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"eip-938\">The expected percentage of the number of pets students have in their homes is distributed (this is the given distribution for the student population of the United States) as in <a class=\"autogenerated-content\" href=\"#fs-idm147763264\">Table 10.12<\/a>.<\/p>\n<div id=\"fs-idm147763264\" class=\"os-table\">\n<table summary=\"Table 10.12\" data-id=\"fs-idm147763264\">\n<thead>\n<tr>\n<th scope=\"col\">Number of Pets<\/th>\n<th scope=\"col\">Percent<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>18<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>25<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>18<\/td>\n<\/tr>\n<tr>\n<td>4+<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.12<\/span><\/div>\n<\/div>\n<p id=\"eip-idp91504704\">A random sample of 1,000 students from the Eastern United States resulted in the data in <a class=\"autogenerated-content\" href=\"#fs-idm64242624\">Table 10.13<\/a>.<\/p>\n<div id=\"fs-idm64242624\" class=\"os-table\">\n<table summary=\"Table 10.13\" data-id=\"fs-idm64242624\">\n<thead>\n<tr>\n<th scope=\"col\">Number of Pets<\/th>\n<th scope=\"col\">Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>210<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>240<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>320<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>140<\/td>\n<\/tr>\n<tr>\n<td>4+<\/td>\n<td>90<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.13<\/span><\/div>\n<\/div>\n<p id=\"eip-idp27106912\">At the 1% significance level, does it appear that the distribution \u201cnumber of pets\u201d of students in the Eastern United States is different from the distribution for the United States student population as a whole? What is the <em data-effect=\"italics\">p<\/em>-value?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"element-866\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><span class=\"os-number\">10.4<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"element-221\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id6230511\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-95\">Suppose you flip two coins 100 times. The results are 20 <em data-effect=\"italics\">HH<\/em>, 27 <em data-effect=\"italics\">HT<\/em>, 30 <em data-effect=\"italics\">TH<\/em>, and 23 <em data-effect=\"italics\">TT<\/em>. Are the coins fair? Test at a 5% significance level.<\/p>\n<\/div>\n<\/div>\n<div id=\"id6230530\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<section class=\"ui-body\" style=\"display: block; overflow: hidden;\" role=\"alert\">\n<h4 data-type=\"solution-title\"><span class=\"os-title-label\">Solution <\/span><span class=\"os-number\">10.4<\/span><\/h4>\n<div class=\"os-solution-container\">\n<p id=\"element-929\">This problem can be set up as a goodness-of-fit problem. The sample space for flipping two fair coins is {<em data-effect=\"italics\">HH<\/em>, <em data-effect=\"italics\">HT<\/em>, <em data-effect=\"italics\">TH<\/em>, <em data-effect=\"italics\">TT<\/em>}. Out of 100 flips, you would expect 25 <em data-effect=\"italics\">HH<\/em>, 25 <em data-effect=\"italics\">HT<\/em>, 25 <em data-effect=\"italics\">TH<\/em>, and 25 <em data-effect=\"italics\">TT<\/em>. This is the expected distribution. The question, &#8220;Are the coins fair?&#8221; is the same as saying, &#8220;Does the distribution of the coins (20 <em data-effect=\"italics\">HH<\/em>, 27 <em data-effect=\"italics\">HT<\/em>, 30 <em data-effect=\"italics\">TH<\/em>, 23 <em data-effect=\"italics\">TT<\/em>) fit the expected distribution?&#8221;<\/p>\n<p id=\"element-570\"><strong>Random Variable:<\/strong> Let <em data-effect=\"italics\">X<\/em> = the number of heads in one flip of the two coins. <em data-effect=\"italics\">X<\/em> takes on the values 0, 1, 2. (There are 0, 1, or 2 heads in the flip of two coins.) Therefore, the <strong>number of cells is three<\/strong>. Since <em data-effect=\"italics\">X<\/em> = the number of heads, the observed frequencies are 20 (for two heads, HH), 57 (for one head, which is the HT and TH counts added together), and 23 (for zero heads or both tails, TT). The expected frequencies are 25 (for two heads), 50 (for one head), and 25 (for zero heads or both tails). This test is right-tailed.<\/p>\n<p id=\"element-774\"><strong><em data-effect=\"italics\">H<sub>0<\/sub><\/em><\/strong>: The coins are fair.<\/p>\n<p id=\"element-734\"><strong><em data-effect=\"italics\">H<sub>1<\/sub><\/em><\/strong>: The coins are not fair.<\/p>\n<p id=\"element-136\"><strong>Distribution for the test:<\/strong><\/p>\n<p>$\\chi_2^2$ where <em data-effect=\"italics\">df<\/em> = 3 \u2013 1 = 2.<\/p>\n<p id=\"element-380\"><strong>Calculate the test statistic:<br \/>\n<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 122px;\">\n<tbody>\n<tr style=\"height: 31px;\">\n<th style=\"width: 10%;\"># of heads<\/th>\n<th style=\"width: 10%; height: 31px;\">$O$<\/th>\n<th style=\"width: 20%; height: 31px;\">$E$<\/th>\n<th style=\"width: 20%; height: 31px;\">$O &#8211; E$<\/th>\n<th style=\"width: 20%; height: 31px;\">$(O &#8211; E)^2$<\/th>\n<th style=\"width: 20%; height: 31px;\">$\\frac{(O &#8211; E)^2}{E}$<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 10%;\">0<\/td>\n<td style=\"width: 10%;\">20<\/td>\n<td style=\"width: 20%; height: 15px;\">25<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:-5}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-5<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:25}\" data-sheets-formula=\"=R[0]C[-1]^2\">25<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10%;\">1<\/td>\n<td style=\"width: 10%;\">57<\/td>\n<td style=\"width: 20%; height: 15px;\">50<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:2}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">7<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:4}\" data-sheets-formula=\"=R[0]C[-1]^2\">49<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:0.16}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.98<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10%;\">2<\/td>\n<td style=\"width: 10%;\">23<\/td>\n<td style=\"width: 20%; height: 15px;\">25<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:5}\" data-sheets-formula=\"=R[0]C[-2]-R[0]C[-1]\">-2<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:25}\" data-sheets-formula=\"=R[0]C[-1]^2\">4<\/td>\n<td style=\"width: 20%;\" data-sheets-value=\"{&quot;1&quot;:3,&quot;3&quot;:1}\" data-sheets-formula=\"=R[0]C[-1]\/R[0]C[-3]\">0.16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> = 2.14<\/p>\n<p id=\"element-490\"><strong>Probability statement:<\/strong> <em data-effect=\"italics\">p<\/em>-value = <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">\u03c7<\/em><sup>2<\/sup> &gt; 2.14) = 0.343 using the Google Sheets function<br \/>\n<code><bdo dir=\"ltr\"><span class=\"formula-content\"><span class=\"default-formula-text-color\" dir=\"auto\">=<\/span><span class=\"default-formula-text-color\" dir=\"auto\">CHISQ.DIST.RT<\/span><span class=\"default-formula-text-color\" dir=\"auto\">(<\/span><span class=\"number\" dir=\"auto\">2.14<\/span><span class=\"default-formula-text-color\" dir=\"auto\">,<\/span>2<span class=\"default-formula-text-color\" dir=\"auto\">)<\/span><\/span><\/bdo><\/code><\/p>\n<p id=\"element-817\"><strong>Compare <em data-effect=\"italics\">\u03b1<\/em> and the <em data-effect=\"italics\">p<\/em>-value:<\/strong><\/p>\n<ul id=\"element-xyz\">\n<li><em data-effect=\"italics\">\u03b1<\/em> = 0.05<\/li>\n<li><em data-effect=\"italics\">p<\/em>-value = 0.343<\/li>\n<\/ul>\n<p id=\"64\"><em data-effect=\"italics\">\u03b1<\/em> &lt; <em data-effect=\"italics\">p<\/em>-value.<\/p>\n<p id=\"element-58\"><strong>Make a decision:<\/strong> Since <em data-effect=\"italics\">\u03b1<\/em> &lt; <em data-effect=\"italics\">p<\/em>-value, do not reject <em data-effect=\"italics\">H<sub>0<\/sub><\/em>.<\/p>\n<p id=\"element-995\"><strong>Conclusion:<\/strong> There is insufficient evidence to conclude that the coins are not fair.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idm153122384\" class=\"statistics try finger ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">Try It <\/span><span class=\"os-number\">10.4<\/span><\/h3>\n<\/header>\n<section>\n<div id=\"eip-860\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-476\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"eip-907\">Students in a social studies class hypothesize that the literacy rates across the world for every region are 82%. <a class=\"autogenerated-content\" href=\"#fs-idm84960960\">Table 10.14<\/a> shows the actual literacy rates across the world broken down by region. What are the test statistic and the degrees of freedom?<\/p>\n<div id=\"fs-idm84960960\" class=\"os-table\">\n<table summary=\"Table 10.14\" data-id=\"fs-idm84960960\">\n<thead>\n<tr>\n<th scope=\"col\">MDG Region<\/th>\n<th scope=\"col\">Adult Literacy Rate (%)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Developed Regions<\/td>\n<td>99.0<\/td>\n<\/tr>\n<tr>\n<td>Commonwealth of Independent States<\/td>\n<td>99.5<\/td>\n<\/tr>\n<tr>\n<td>Northern Africa<\/td>\n<td>67.3<\/td>\n<\/tr>\n<tr>\n<td>Sub-Saharan Africa<\/td>\n<td>62.5<\/td>\n<\/tr>\n<tr>\n<td>Latin America and the Caribbean<\/td>\n<td>91.0<\/td>\n<\/tr>\n<tr>\n<td>Eastern Asia<\/td>\n<td>93.8<\/td>\n<\/tr>\n<tr>\n<td>Southern Asia<\/td>\n<td>61.9<\/td>\n<\/tr>\n<tr>\n<td>South-Eastern Asia<\/td>\n<td>91.9<\/td>\n<\/tr>\n<tr>\n<td>Western Asia<\/td>\n<td>84.5<\/td>\n<\/tr>\n<tr>\n<td>Oceania<\/td>\n<td>66.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><span class=\"os-number\">10.14<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":1,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-94","chapter","type-chapter","status-publish","hentry"],"part":89,"_links":{"self":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/94","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":9,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/94\/revisions"}],"predecessor-version":[{"id":776,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/94\/revisions\/776"}],"part":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/parts\/89"}],"metadata":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/94\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/media?parent=94"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=94"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/contributor?post=94"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/license?post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}