{"id":33718,"date":"2021-10-07T18:37:17","date_gmt":"2021-10-07T18:37:17","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=33718"},"modified":"2026-03-01T08:19:39","modified_gmt":"2026-03-01T08:19:39","slug":"testing-the-independence-based-on-contingency-table-data","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/testing-the-independence-based-on-contingency-table-data\/","title":{"rendered":"Testing Independence based on Contingency Table Data"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3.png\",\n  \"caption\": \"CFA Level I Chi-Square Table\",\n  \"width\": 445,\n  \"height\": 427,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/twTTWMuEGx8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>To test the relationship between two categorical variables, we use a contingency table and a test of independence based on a chi-square distribution. The test statistic is calculated as follows:<\/p>\n<p>$$<br>\\chi^{2}=\\sum_{i=1}^{m} \\frac{\\left(O_{i j}-E_{i j}\\right)^{2}}{E_{i j}}<br>$$<\/p>\n<p>Where :<\/p>\n<p>\\(E_{i j}=\\frac{(\\text { Total row } i) \\times(\\text { Total column } j)}{\\text { Overall total }}\\)<\/p>\n<p>\\(m\\) = Number of cells in the table, the number of groups in the first class, multiplied by the number of groups in the second class.<\/p>\n<p>\\(O_{i j}\\) = Number of observations in each cell of row <em>i <\/em>and column <em>j <\/em>(i.e., observed frequency).<\/p>\n<p>\\(E_{i j}\\) = Expected number of observations in each cell of row <em>i <\/em>and column <em>j<\/em>, assuming independence (i.e., expected frequency).<\/p>\n<p>The degrees of freedom is given by:<\/p>\n<p>$$\\text{Degrees of freedom} = (r-1)(c-1)$$<\/p>\n<p>Where:<\/p>\n<p>\\(r\\)<em> = <\/em>Number of rows.<\/p>\n<p>\\(c\\) = Number of columns.<\/p>\n<p><strong>Example: Testing Independence Based on Contingency Table Data<\/strong><\/p>\n<p>The following contingency table shows the responses of two categories of investors (employed vs. retired) with regard to their primary investment objectives (growth, income, or both). The total sample size is 173.<\/p>\n<p>$$<br>\\begin{array}{l|c|c|c|c}&nbsp;<br>{} &amp; \\textbf { Growth } &amp; \\textbf { Income } &amp; \\textbf { Both } &amp; \\textbf { Total } \\\\<br>\\hline \\text { Employed } &amp; 52 &amp; 25 &amp; 10 &amp; 87 \\\\<br>\\hline \\text { Retired } &amp; 32 &amp; 47 &amp; 7 &amp; 86 \\\\<br>\\hline\\textbf{ Total } &amp; 84 &amp; 72 &amp; 17 &amp; 173 \\\\<br>\\end{array}<br>$$<\/p>\n<p>Use a 95% significance level to test whether there is any significant difference between employed and retired investors with regard to primary investment objectives. &nbsp;&nbsp;<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>\\(H_0\\): There is no significant difference between employed and retired investors with regard to primary investment objectives.<\/p>\n<p>\\(H_\u03b1\\): There is a significant difference between employed and retired investors with regard to primary investment objectives.<\/p>\n<p><strong><em>Step 1:<\/em> <\/strong>We calculate the expected frequency of investors by their category (employed vs. retired) and investment objective using the following formula:<\/p>\n<p>$$ E_{i j}=\\frac{(\\text { Total row } i) \\times(\\text { Total column } j)}{\\text { Overall total }} $$<\/p>\n<p>$$\\begin{array}{l|c|c|c}<br>{} &amp; \\textbf { Growth } &amp; \\textbf { Income } &amp; \\textbf { Both } \\\\<br>\\hline \\text { Employed } &amp; \\frac{\\left(87\\times 84\\right)}{173}=42 &amp; \\frac{\\left(87\\times 72\\right)}{ 173}=36 &amp; \\frac{\\left(87\\times 17\\right)}{173}=9 \\\\<br>\\hline \\text { Retired } &amp; \\frac{\\left(86\\times 84\\right)}{173}=42 &amp; \\frac{\\left(86\\times 72\\right)}{173}=36 &amp; \\frac{\\left(86\\times 17\\right)}{ 173}=8 \\\\ \\hline \\textbf { Total } &amp; \\mathbf{8 4} &amp; \\mathbf{7 2} &amp; \\mathbf{1 7} \\\\<br>&nbsp;\\end{array} $$<\/p>\n<p><b><strong><em>Step 2: <\/em><\/strong><\/b>We calculate the scaled squared deviation for each combination of investor category and investment objective as follows:<\/p>\n<p>$$ \\begin{array}{l|c|c|c}<br>&nbsp;&amp; \\textbf { Growth } &amp; \\textbf { Income } &amp; \\textbf { Both } \\\\<br>\\hline \\text { Employed } &amp; \\frac{(52-42)^{2}}{ 42}=2.254 &amp; \\frac{(25-36)^ 2}{ 36}=0.469 &amp; \\frac{(10-9)^2}{9}=0.246 \\\\<br>\\hline \\text { Retired } &amp; \\frac{(32-42)^{2}}{42}=2.280 &amp; \\frac{(47-36)^2}{36}=3.510 &amp; \\frac{(7-8)^2}{8}=0.249\\\\<br>\\hline \\textbf { Total } &amp; 4.534 &amp; 6.979 &amp; 0.495 \\\\<br>\\end{array} $$<\/p>\n<p><em><strong>Step 3:<\/strong><\/em> We calculate the value of \\(X^2&nbsp; = 4.532 + 6.976 + 0.494 = 12.002\\).<\/p>\n<p>$$ \\chi^2=4.534+6.979+0.495=12.008 $$&nbsp;<\/p>\n<p><em><strong>Step 4:<\/strong> <\/em>The critical value of \\(X^2\\)<b>&nbsp;<\/b>is 5.99. It is determined using the following:<\/p>\n<ul>\n<li>There are \\((r \u2212 1)(c \u2212 1) = (2 \u2212 1) \u00d7 (3 \u2212 1) = 2\\) degrees of freedom.<\/li>\n<li>It is a one-sided test with a 5% level of significance.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-38717\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3.png\" alt=\"\" width=\"445\" height=\"427\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3.png 445w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3-300x288.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3-400x384.png 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/10\/cfa-level-1-chi-square-table-3-24x24.png 24w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><\/p>\n<p><em><strong>Decision rule:<\/strong> <\/em>The calculated value of \\(\\chi^2 =12.008\\) is greater than the critical value of 5.99. As such, there is sufficient evidence to support the conclusion that retired investors and employed investors have different primary investment objectives.&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>To test the relationship between two categorical variables, we use a contingency table and a test of independence based on a chi-square distribution. The test statistic is calculated as follows: $$\\chi^{2}=\\sum_{i=1}^{m} \\frac{\\left(O_{i j}-E_{i j}\\right)^{2}}{E_{i j}}$$ Where : \\(E_{i j}=\\frac{(\\text {&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-33718","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Testing Independence with Contingency Tables | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to test the relationship between categorical variables using contingency tables and chi-square analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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