{"id":134,"date":"2018-03-14T10:45:00","date_gmt":"2018-03-14T09:45:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/2018\/03\/14\/chi-square-test-for-independence-question-17a-research-team\/"},"modified":"2026-07-13T00:29:08","modified_gmt":"2026-07-12T22:29:08","slug":"chi-square-test-for-independence-question-17a-research-team","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/chi-square-test-for-independence-question-17a-research-team\/","title":{"rendered":"Chi-Square Test for Independence \u2013 Question 17(A research team\u2026)"},"content":{"rendered":"<p><!-- ktg-updated-banner --><\/p>\n<div class=\"ktg-updated-banner\" style=\"margin:1em 0;padding:0.75em 1em;background:#eff6ff;border-left:4px solid #3b82f6;border-radius:4px;\">\n<p><strong>Updated July 12, 2026:<\/strong> Refreshed for SEO \u2014 Question 17 (Chi-square independence). Part of the <a href=\"https:\/\/kindsonthegenius.com\/blog\/hypothesis-testing-solved-examplesquestions-and-solutions\/\">20 hypothesis testing solved examples<\/a> series.<\/p>\n<\/div>\n<nav class=\"ktg-question-nav\" aria-label=\"Hypothesis testing question navigation\" style=\"margin:1em 0;padding:0.75em 1em;background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px;display:flex;flex-wrap:wrap;gap:1em;justify-content:space-between;\"><a href=\"https:\/\/kindsonthegenius.com\/blog\/chi-square-goodness-of-fit-test-question-16-a-newly-developed-muesli\/\">\u2190 Question 16<\/a> \u00b7 <a href=\"https:\/\/kindsonthegenius.com\/blog\/hypothesis-testing-solved-examplesquestions-and-solutions\/\">All 20 questions<\/a> \u00b7 <a href=\"https:\/\/kindsonthegenius.com\/blog\/chi-square-test-for-independence-question-18-a-publisher-is-interested\/\">Question 18 \u2192<\/a><\/nav>\n<p><span style=\"color: #cc0000;\"><span style=\"font-size: large;\"><b>Question 17<\/b><\/span><\/span><br \/>\nA research team investigated whether there was any significant correlation between the severity of a certain disease runoff and the age of the patients. During the study, data for n = 200 patients were collected and grouped according to the severity of the disease and the age of the patient. The table below shows the result<\/p>\n<p>&nbsp;<\/p>\n<table style=\"height: 172px; width: 402px;\" border=\"1\">\n<tbody>\n<tr>\n<td width=\"62\"><\/td>\n<td width=\"63\"><\/td>\n<td colspan=\"3\" align=\"center\"><b>Age<\/b><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td width=\"69\"><b>below 40<\/b><\/td>\n<td width=\"55\"><b>40 &#8211; 60<\/b><\/td>\n<td width=\"66\"><b>above 60<\/b><\/td>\n<\/tr>\n<tr>\n<td rowspan=\"3\"><b>runoff<\/b><\/td>\n<td><b>slight<\/b><\/td>\n<td>41<\/td>\n<td>34<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td><b>average<\/b><\/td>\n<td>25<\/td>\n<td>25<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td><b>serious<\/b><\/td>\n<td>6<\/td>\n<td>33<\/td>\n<td>15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Let us decide about the correlation between the age of the patients and the severity of disease progression.<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #38761d;\"><span style=\"font-size: large;\"><b>Solution Steps<\/b><\/span><\/span><br \/>\nAs usual, we need to understand the problem and decide on which particular test to carry out.<\/p>\n<p>In this case, since the question says to investigate whethere there was any significant correlation between the severity and age, it means that the null hypothesis would be that &#8216;there is correlation between the age and the severity&#8217;. That is the hypothesis we are going to test.<\/p>\n<p>&nbsp;<\/p>\n<p><b>Step 1: State the null and alternate hypothesis<\/b><br \/>\n<b>H<sub>0<\/sub>:<\/b> there is significant correlation between the severity and the age<br \/>\n<b>H<sub>a<\/sub><\/b>: there is no significant correlation between the severity and the age<\/p>\n<p>&nbsp;<\/p>\n<p>Since we are going to be using Excel to simplify the solving of this problem, I have transfered the table to MS Excel. This is shown in Table 1. <a href=\"https:\/\/kindsonthegenius.com\/blog\/statistical-tables\/\" target=\"_blank\" rel=\"noopener noreferrer\">You can get the completed excel sheet from here<\/a><\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-TuFa-9YBPbA\/Wqjqs-WBkkI\/AAAAAAAABT0\/samb9vx-y203HGnsQmgsVIQ3RKKZICWiACLcBGAs\/s1600\/Chi-Square-Question-17-Table1.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-TuFa-9YBPbA\/Wqjqs-WBkkI\/AAAAAAAABT0\/samb9vx-y203HGnsQmgsVIQ3RKKZICWiACLcBGAs\/s400\/Chi-Square-Question-17-Table1.jpg\" width=\"400\" height=\"152\" border=\"0\" data-original-height=\"382\" data-original-width=\"1001\" \/>\u00a0<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 1<\/div>\n<p>&nbsp;<\/p>\n<p><b><br \/>\n<\/b><br \/>\n<b>Step 2: Calcualte the totals<\/b><br \/>\nIn this step we calculate the totals for each of the row. This i have done using excel formula as you can see in Table 2<\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-YZ3TdhunhDc\/WqjrqIN8UdI\/AAAAAAAABUI\/VNP3-Hj4MoUT727MkPFnQS56Mf9dX0hvgCLcBGAs\/s1600\/Chi-Square-Question-17-Table2.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-YZ3TdhunhDc\/WqjrqIN8UdI\/AAAAAAAABUI\/VNP3-Hj4MoUT727MkPFnQS56Mf9dX0hvgCLcBGAs\/s400\/Chi-Square-Question-17-Table2.jpg\" width=\"400\" height=\"146\" border=\"0\" data-original-height=\"444\" data-original-width=\"1203\" \/><\/a><\/div>\n<div style=\"clear: both; text-align: center;\"><\/div>\n<div style=\"clear: both; text-align: center;\">Table 2<\/div>\n<p>&nbsp;<\/p>\n<p><b>Step 3: Calculate the expected values<\/b><br \/>\nThe expected values are calculated by multiplying the corresponding row and column sub-total and dividing\u00a0 by the grand-total. For example, the first expected value that corresponds to Slight and Below 40 would be calculated as follows:<\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/2.bp.blogspot.com\/-7ioGG-pEbQM\/WqjtD-juCAI\/AAAAAAAABUY\/l0ARwRERgjsTELogKhPZXg3jn9dRC00tgCLcBGAs\/s1600\/Question%2B17-%2BSingle-Expected.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/2.bp.blogspot.com\/-7ioGG-pEbQM\/WqjtD-juCAI\/AAAAAAAABUY\/l0ARwRERgjsTELogKhPZXg3jn9dRC00tgCLcBGAs\/s200\/Question%2B17-%2BSingle-Expected.jpg\" width=\"200\" height=\"57\" border=\"0\" data-original-height=\"82\" data-original-width=\"283\" \/><\/a><\/div>\n<div style=\"clear: both; text-align: center;\"><\/div>\n<p>Do this for all the 9 observed values. I have used excel to automatically generate these values and it is shown in Table 3<\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/1.bp.blogspot.com\/-2r2n0yImdTQ\/Wqjv6TT9w8I\/AAAAAAAABUo\/5Xc5U0hO6cgzcBZAMN19hockwhCIHpQPwCLcBGAs\/s1600\/Chi-Square-Question-17-Table3.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-2r2n0yImdTQ\/Wqjv6TT9w8I\/AAAAAAAABUo\/5Xc5U0hO6cgzcBZAMN19hockwhCIHpQPwCLcBGAs\/s400\/Chi-Square-Question-17-Table3.jpg\" width=\"400\" height=\"163\" border=\"0\" data-original-height=\"271\" data-original-width=\"660\" \/>\u00a0<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 3<\/div>\n<p>&nbsp;<\/p>\n<p><b>Step 4: Calculate Squared Difference (O-E)2<\/b><br \/>\nWhere O is the observed values in Table 2 and E is the expected values calcualted in Table 3. The first squared difference would be.<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-GCBj1oOuh1o\/Wqjw4lIbY6I\/AAAAAAAABU0\/azMYGH42mf0GjxThTWuysPZ4_nHol0P1QCLcBGAs\/s1600\/Chi-square-Squared-Deviation-Single.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-GCBj1oOuh1o\/Wqjw4lIbY6I\/AAAAAAAABU0\/azMYGH42mf0GjxThTWuysPZ4_nHol0P1QCLcBGAs\/s320\/Chi-square-Squared-Deviation-Single.jpg\" width=\"320\" height=\"49\" border=\"0\" data-original-height=\"61\" data-original-width=\"382\" \/><\/a><\/div>\n<p>&nbsp;<\/p>\n<p>Do this for all the the observed values and the corresponding expected values. The resulting sets of values is given in Table 4<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-SeddBv2Rwvg\/WqjzJ20YxsI\/AAAAAAAABVA\/KZHFZe8YnTI9tE_yWmoAzzU0KOlPV_VlgCLcBGAs\/s1600\/Chi-Square-Question-17-Table4.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-SeddBv2Rwvg\/WqjzJ20YxsI\/AAAAAAAABVA\/KZHFZe8YnTI9tE_yWmoAzzU0KOlPV_VlgCLcBGAs\/s400\/Chi-Square-Question-17-Table4.jpg\" width=\"400\" height=\"163\" border=\"0\" data-original-height=\"401\" data-original-width=\"973\" \/>\u00a0<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 4<\/div>\n<p>&nbsp;<\/p>\n<p><b>Step 5: Calculate the Component<\/b><br \/>\nThis is the squared deviation you calculated in step 4 divided by the corresponding expected values. For the first value it would be<\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/1.bp.blogspot.com\/-roa2BXCaeCY\/Wqj0NIEeLCI\/AAAAAAAABVM\/ZY0vtC_vmn8l9oPQm2MEclZf51xogPfOACLcBGAs\/s1600\/Q17%2BChi-Square-Single%2BCalculation.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-roa2BXCaeCY\/Wqj0NIEeLCI\/AAAAAAAABVM\/ZY0vtC_vmn8l9oPQm2MEclZf51xogPfOACLcBGAs\/s200\/Q17%2BChi-Square-Single%2BCalculation.jpg\" width=\"200\" height=\"63\" border=\"0\" data-original-height=\"83\" data-original-width=\"260\" \/><\/a><\/div>\n<p>&nbsp;<\/p>\n<p>If you repeat this all the values, then the resulting table would be table 5.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-I8PUakWpsLw\/Wqj1V8_A1CI\/AAAAAAAABVY\/iXikvaz7omYKUps7fsKU-33eHagQIlPBwCLcBGAs\/s1600\/Chi-Square-Question-17-Table5.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-I8PUakWpsLw\/Wqj1V8_A1CI\/AAAAAAAABVY\/iXikvaz7omYKUps7fsKU-33eHagQIlPBwCLcBGAs\/s400\/Chi-Square-Question-17-Table5.jpg\" width=\"400\" height=\"215\" border=\"0\" data-original-height=\"555\" data-original-width=\"1031\" \/><\/a><\/div>\n<p>&nbsp;<\/p>\n<p><b>Step 6: Calculate the Test Statistic<\/b><\/p>\n<p>This is the sum of all the terms in calculated in the table. I calculated this using the Sum() formula in Excel, but you can do this by hand just to verify.<\/p>\n<p>Test Statistic = 3.83 + 0.56 + 2.48 + 0.32 + 0.43 + 0.06 + 9.29 +2.68 + 2.87 <b>= 22.52<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><b>Step 7: Look up the critical Value from Chi-Square table<\/b><br \/>\n<a href=\"http:\/\/kindsonthegenius.blogspot.com\/2018\/03\/statistical-tables.html\" target=\"_blank\" rel=\"noopener noreferrer\">Get Statistical table from here <\/a><br \/>\nFirst we calcuale the degrees of freedom<br \/>\ndf = (3-1) * (3-1) =\u00a0 4<br \/>\nalpha = 0.01<\/p>\n<p>The critical value from the table of Chi-Square distribution is written as<\/p>\n<p>K<sub>0.01, 4<\/sub> = <b>13.28<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><b>Step 8: State your conclusion<\/b><br \/>\nSince the calculated value of the test statistic is greater than the critical value, we therefore reject the null hypothesis and conclude that the data is not related.<\/p>\n<p>The whole tables are shown below, you can also <a href=\"https:\/\/kindsonthegenius.com\/blog\/statistical-tables\/\" target=\"_blank\" rel=\"noopener noreferrer\">download it for free<\/a><\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/2.bp.blogspot.com\/-4QTSNxxMVeA\/Wqj7fNUdV7I\/AAAAAAAABVo\/4_PR0c2jsCc13WmOXaeADQN4R2NiNOgPwCLcBGAs\/s1600\/Chi-Square-Question%2B17%2Ball%2Btables.jpg\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/2.bp.blogspot.com\/-4QTSNxxMVeA\/Wqj7fNUdV7I\/AAAAAAAABVo\/4_PR0c2jsCc13WmOXaeADQN4R2NiNOgPwCLcBGAs\/s640\/Chi-Square-Question%2B17%2Ball%2Btables.jpg\" width=\"640\" height=\"332\" border=\"0\" data-original-height=\"631\" data-original-width=\"1203\" \/><\/a><\/div>\n<p><a href=\"https:\/\/kindsonthegenius.com\/blog\/statistical-tables\/\" target=\"_blank\" rel=\"noopener noreferrer\">Download this excel sheet for free<\/a><\/p>\n<section class=\"ktg-related\" style=\"margin:2em 0;padding:1em 1.25em;background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px;\">\n<h2>Related<\/h2>\n<ul>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/hypothesis-testing-solved-examplesquestions-and-solutions\/\">All hypothesis testing solved examples<\/a><\/li>\n<li><strong>Test type:<\/strong> Chi-square independence<\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/chi-square-goodness-of-fit-example-step-by-step-procedure\/\">Method tutorial<\/a><\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/statistics-tutorial-3-hypothesis-testing\/\">Hypothesis testing basics<\/a><\/li>\n<\/ul>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>Updated July 12, 2026: Refreshed for SEO \u2014 Question 17 (Chi-square independence). Part of the 20 hypothesis testing solved examples series. \u2190 Question 16 \u00b7 &hellip; <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[223],"tags":[],"class_list":["post-134","post","type-post","status-publish","format-standard","hentry","category-hypothesis-testing"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/134","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=134"}],"version-history":[{"count":5,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/134\/revisions"}],"predecessor-version":[{"id":2234,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/134\/revisions\/2234"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}