{"id":137,"date":"2018-03-13T04:23:00","date_gmt":"2018-03-13T03:23:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/2018\/03\/13\/question-14-chi-square-goodness-of-fit-test-problem-can-a-dice-be-considered\/"},"modified":"2026-07-13T00:28:34","modified_gmt":"2026-07-12T22:28:34","slug":"question-14-chi-square-goodness-of-fit-test-problem-can-a-dice-be-considered","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/question-14-chi-square-goodness-of-fit-test-problem-can-a-dice-be-considered\/","title":{"rendered":"Question 14 \u2013 Chi-Square Goodness of Fit Test Problem( Can a dice be considered\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 14 (Chi-square goodness of fit). 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\/question-13-chi-square-test-for-independence-step-by-step-procedurepolitical-affilication\/\">\u2190 Question 13<\/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\/how-to-perform-mann-witney-u-teststep-by-step-hypothesis-testing\/\">Question 15 \u2192<\/a><\/nav>\n<div style=\"color: #555555; font-size: 18px; line-height: 30px; text-align: justify;\">\n<div style=\"font-family: 'segoe ui';\"><span style=\"color: #cc0000;\"><span style=\"font-size: large;\"><b>Question 14<\/b><\/span><\/span><\/p>\n<hr \/>\n<p>Can a dice be considered regular which is showing the following frequency distribution during 1000 throws?<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/2.bp.blogspot.com\/-u6X7UpDojv0\/WqdBfn2sdII\/AAAAAAAABSE\/Fvsh37kNX4oXgS2wr_HFZRKjeyf64Cf4gCLcBGAs\/s1600\/Dice.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"296\" data-original-width=\"295\" height=\"140\" src=\"https:\/\/2.bp.blogspot.com\/-u6X7UpDojv0\/WqdBfn2sdII\/AAAAAAAABSE\/Fvsh37kNX4oXgS2wr_HFZRKjeyf64Cf4gCLcBGAs\/s200\/Dice.jpg\" width=\"140\" \/><\/a><\/div>\n<table border=\"1\" style=\"height: 70px; width: 399px;\">\n<tbody>\n<tr>\n<td><b>Thrown Value<\/b><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td><b>Frequency<\/b><\/td>\n<td>182<\/td>\n<td>154<\/td>\n<td>162<\/td>\n<td>175<\/td>\n<td>151<\/td>\n<td>176<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #38761d;\"><span style=\"font-size: large;\"><b>Solution Steps&nbsp;<\/b><\/span><\/span><\/p>\n<hr \/>\n<p>In this case, the questions says to consider whether the dice can be considered regular(consistent). So we use Chi-Square goodness of fit test which is used to determine whether a smple data is consistent with a distribution.<\/p>\n<p><a href=\"https:\/\/youtu.be\/pXny5vZwoaI\" target=\"_blank\" rel=\"noopener\">Watch the video explanation here<\/a><br \/><a href=\"http:\/\/kindsonthegenius.blogspot.com\/2018\/03\/excel-sheets-for-statistics-with.html\" target=\"_blank\" rel=\"noopener\">Get the excel sheet for this question <\/a><\/p>\n<p>The formular for chi-square is<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/4.bp.blogspot.com\/-1ojqbmJPLeI\/WqXYikGVsZI\/AAAAAAAABPI\/nMwJvbP9ERAvrxR1-wpaO5k4fPMuiRy6wCPcBGAYYCw\/s1600\/Chi-Square-Formular.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"103\" data-original-width=\"287\" height=\"71\" src=\"https:\/\/4.bp.blogspot.com\/-1ojqbmJPLeI\/WqXYikGVsZI\/AAAAAAAABPI\/nMwJvbP9ERAvrxR1-wpaO5k4fPMuiRy6wCPcBGAYYCw\/s200\/Chi-Square-Formular.jpg\" width=\"200\" \/><\/a><\/div>\n<p>where<br \/>O = Observed values<br \/>E = Expected value <\/p>\n<p><b>Step 1: State the Null and althernate hypothesis<\/b><\/p>\n<p><b>H<sub>0<\/sub>:<\/b> the data is consistent with a specified distribution<br \/><b>H<sub>a<\/sub><\/b>: the data is not consistent with a specified distributions<\/p>\n<p><b>Step 2:&nbsp; Create the Table<\/b><br \/>I have tranfered the data to excel and created the table as shown in Table 1.<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/3.bp.blogspot.com\/-ZDjD3zrKRqU\/WqdMOHW5QwI\/AAAAAAAABSU\/ZP8ypTAY900oSSjRjNl_kKI5N7r5wAIzACLcBGAs\/s1600\/Question%2B14-Table1.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"607\" data-original-width=\"770\" height=\"252\" src=\"https:\/\/3.bp.blogspot.com\/-ZDjD3zrKRqU\/WqdMOHW5QwI\/AAAAAAAABSU\/ZP8ypTAY900oSSjRjNl_kKI5N7r5wAIzACLcBGAs\/s320\/Question%2B14-Table1.jpg\" width=\"320\" \/>&nbsp;<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 1 <\/div>\n<p><ins data-ad-client=\"ca-pub-7041870931346451\" data-ad-format=\"fluid\" data-ad-layout=\"in-article\" data-ad-slot=\"4209786523\" style=\"display: block; text-align: center;\"><\/ins> <br \/><b>Step 3: Calcualte the Expected Value (E)<\/b><br \/>The expected value is the same as the mean of the sample which I have calculated as shown in the table.<\/p>\n<p>E = 166.67<\/p>\n<p><b>Step 4: Calculated the Differences<\/b><br \/>In this step, you need to calculate the Observed Value &#8211; Expected Value (O-E) for each of the obseravtion. This I have done and added additional column to the table.<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/4.bp.blogspot.com\/-cFSpRC2ve0o\/WqdNYua5OzI\/AAAAAAAABSg\/7GcPrRcW7NIm11D_qGLCqYyJL7CTeEAiwCLcBGAs\/s1600\/Question%2B14-Table2.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"614\" data-original-width=\"914\" height=\"214\" src=\"https:\/\/4.bp.blogspot.com\/-cFSpRC2ve0o\/WqdNYua5OzI\/AAAAAAAABSg\/7GcPrRcW7NIm11D_qGLCqYyJL7CTeEAiwCLcBGAs\/s320\/Question%2B14-Table2.jpg\" width=\"320\" \/>&nbsp;<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 2 <\/div>\n<p><b>Step 5: Calculate the Squared Differences<\/b><br \/>This is simple the square of the differences calculated in Step 4. I have added another column to our table to hold the squared differences.<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/3.bp.blogspot.com\/-fwYxewStGUg\/WqdOFLp6shI\/AAAAAAAABSo\/09ucdmZrsiUTjb1-5zCWYHyQIIncEirIQCLcBGAs\/s1600\/Question%2B14-Table3.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"561\" data-original-width=\"930\" height=\"193\" src=\"https:\/\/3.bp.blogspot.com\/-fwYxewStGUg\/WqdOFLp6shI\/AAAAAAAABSo\/09ucdmZrsiUTjb1-5zCWYHyQIIncEirIQCLcBGAs\/s320\/Question%2B14-Table3.jpg\" width=\"320\" \/>&nbsp;<\/a><\/div>\n<div style=\"clear: both; text-align: center;\">Table 3 <\/div>\n<p><b>Step 6: Calculate the Component<\/b><br \/>This is calculated by dividing the squared difference calculated in step 5 by the expected value for each of the value.<br \/>Our table has been updated to include this column as well as the sum of this column<\/p>\n<div style=\"clear: both; text-align: center;\"><a href=\"https:\/\/4.bp.blogspot.com\/-uDO04jc9x-g\/Wqfe8K7Xy3I\/AAAAAAAABTU\/K-c9FsOkhHwth8IetY6QtCG15Cii7ZAzgCLcBGAs\/s1600\/Question%2B14-Table4-updated.jpg\" style=\"margin-left: 1em; margin-right: 1em;\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" border=\"0\" data-original-height=\"414\" data-original-width=\"1245\" height=\"211\" src=\"https:\/\/4.bp.blogspot.com\/-uDO04jc9x-g\/Wqfe8K7Xy3I\/AAAAAAAABTU\/K-c9FsOkhHwth8IetY6QtCG15Cii7ZAzgCLcBGAs\/s640\/Question%2B14-Table4-updated.jpg\" width=\"640\" \/><\/a><\/div>\n<div style=\"clear: both; text-align: center;\"><\/div>\n<div style=\"clear: both; text-align: center;\">Table 4 <\/div>\n<p><b>Step 7: Calculate the Chi-Square Statistic<\/b><br \/>The chi-square statistic is calculated as the sum of the last column in our table which is the same as the result of using the formula<\/p>\n<p>Chi-Square Statistic = 4.92<\/p>\n<p><b>Step 8: Find the P-Value from Tables<\/b><br \/>You can find the p-value using the calculate chi-square statistic and the degrees of freedom<\/p>\n<p>Degrees of freedom is given by N-1<br \/>df = 6 &#8211; 1 =5<br \/>where N is the number of observation. In this case <\/p>\n<p><a href=\"http:\/\/kindsonthegenius.blogspot.com\/2018\/03\/statistical-tables.html\" target=\"_blank\" rel=\"noopener\">See statistical tables here <\/a><\/p>\n<p>From the chi-square table we for <span lang=\"el\">\u03b1 = 0.05, df = 5<\/span><br \/>get a p-value of&nbsp; <b>11.07<\/b><\/p>\n<p><b>Step 9: State your Conclusion<\/b><br \/>In this case&nbsp; the calculated test statistic from the sample which is 4.92 is less than the critical value from the table which is 11.07.<br \/>Therefore we accept the null hypothesis<\/p>\n<p><a href=\"http:\/\/kindsonthegenius.blogspot.com\/2018\/03\/excel-sheets-for-statistics-with.html\" target=\"_blank\" rel=\"noopener\">Get this excel file<\/a><\/p>\n<\/div>\n<\/div>\n<p><ins data-ad-client=\"ca-pub-7041870931346451\" data-ad-format=\"fluid\" data-ad-layout=\"in-article\" data-ad-slot=\"4209786523\" style=\"display: block; text-align: center;\"><\/ins> <br \/><ins data-ad-client=\"ca-pub-7041870931346451\" data-ad-format=\"fluid\" data-ad-layout=\"in-article\" data-ad-slot=\"4209786523\" style=\"display: block; text-align: center;\"><\/ins><\/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 goodness of fit<\/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 14 (Chi-square goodness of fit). Part of the 20 hypothesis testing solved examples series. \u2190 Question &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-137","post","type-post","status-publish","format-standard","hentry","category-hypothesis-testing"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/137","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=137"}],"version-history":[{"count":2,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/137\/revisions"}],"predecessor-version":[{"id":2231,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/137\/revisions\/2231"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}