{"id":11786,"date":"2021-03-03T13:22:23","date_gmt":"2021-03-03T13:22:23","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11786"},"modified":"2026-02-13T20:04:33","modified_gmt":"2026-02-13T20:04:33","slug":"hypothesis-testing-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/hypothesis-testing-2\/","title":{"rendered":"Hypothesis Testing"},"content":{"rendered":"<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8?si=9V3HUzUHTNKgS2E5\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Multiple Regression (2022 Level II CFA\u00ae Exam \u2013 Reading 2)\",\n  \"description\": \"In this video, Professor James Forjan, PhD, CFA, explains multiple regression for the CFA Level II Quantitative Methods topic. The lesson builds on simple linear regression and shows how to evaluate multiple independent variables, interpret regression coefficients, analyze p-values, calculate F-statistics, test hypotheses, and apply regression models in practical financial analysis.\",\n  \"uploadDate\": \"2021-12-23\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/8E2AbtAb0a8\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=8E2AbtAb0a8\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8\",\n  \"duration\": \"PT55M42S\",\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/wp-content\/uploads\/2023\/01\/analystprep-logo.png\"\n    }\n  }\n}\n<\/script><\/p>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test.png\",\n  \"caption\": \"Rejection Regions for a Two-Sided Hypothesis Test\",\n  \"width\": 1242,\n  \"height\": 961,\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><br \/>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-t-table-2.36.png\",\n  \"caption\": \"t-Distribution Critical Values Table\",\n  \"width\": 750,\n  \"height\": 720,\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><\/p>\n<p>Hypothesis testing involves testing an assumption regarding a population parameter. A null hypothesis is a condition believed not to be true. We reject the null hypothesis in the presence of enough evidence against it and accept the alternative hypothesis.<\/p>\n<p>Hypothesis testing is performed on the estimated slope coefficients to establish if the independent variables make a significant contribution to explaining the variation in the dependent variable.<\/p>\n<p><!-- TOP CTA \u2013 Full Width Outline Button --><\/p>\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:13px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:17px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \"><br \/>\n     Practice advanced hypothesis testing with free trial access.<br \/>\n  <\/a>\n<\/div>\n<p>The t-statistic for testing the significance of the individual coefficients in a multiple regression model is calculated using the formula:<\/p>\n<p>$$t=\\frac{\\widehat{b_{j}}-b_{H0}}{S_{\\widehat{b_{j}}}}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(\\widehat{b_{j}}\\) = Estimated regression coefficient.<\/li>\n<li data-tadv-p=\"keep\">\\(b_{H0}\\) = Hypothesized value.<\/li>\n<li data-tadv-p=\"keep\">\\(S_{\\widehat{b_{j}}}\\) The standard error of the estimated coefficient.<\/li>\n<\/ul>\n<p>Note that the test statistic has \\(n-k-1\\) degrees of freedom, where \\(k\\) is the number of independent variables, and \\(1\\) is the intercept term.<\/p>\n<p>A t-test tests the null hypothesis that the regression coefficient is equal to some hypothesized value against the alternative hypothesis that it is not.<\/p>\n<p style=\"text-align: center;\">\\(H_0:b_j=v\\) v.s \\(H_{a}:b_{j}\u2260v\\)<\/p>\n<p>Where:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">\\(v\\) = Hypothesized value.<\/li>\n<\/ul>\n<p>The following are the steps followed in hypothesis testing:<\/p>\n<p><strong>\u00a0 i. Formulate hypothesis.<\/strong><\/p>\n<ul>\n<li>For example,\u00a0<em>Null hypothesis, \\(H_0: b_{j}=0\\) vs Alternative hypothesis, \\(H_{a}: b_{j}\u22600\\).<\/em><\/li>\n<\/ul>\n<p><strong>\u00a0 ii. Determine the suitable test statistic.<\/strong><\/p>\n<p>$$t_{statistic}=\u00a0\\frac{\\widehat{b_{j}}-b_{H0}}{S_{\\widehat{b_{j}}}}$$<\/p>\n<p><strong>\u00a0 \u00a0iii. Specify the significance level.<\/strong><\/p>\n<ul>\n<li>5% significance level is commonly used.<\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0iv. Make the statistical decision.<\/strong><\/p>\n<ul>\n<li>Reject the null hypothesis when the t-statistic is greater than the absolute value of the critical value of t.<\/li>\n<\/ul>\n<h4>Example: The Test of Significance for a Regression Coefficient<\/h4>\n<p>Consider the following multiple regression results of the price of USDX versur inflation rate and real interest rate.<\/p>\n<p>$$\\small{\\begin{array}{l|c}\\hline\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.8264\\\\ \\hline\\text{R Square} &amp; 0.6830\\\\ \\hline\\text{Adjusted R Square} &amp; 0.5924\\\\ \\hline\\text{Standard Error} &amp; 5.3537\\\\ \\hline\\text{Observations} &amp; 10\\\\ \\hline\\end{array}}$$<\/p>\n<p>$$\\small{\\begin{array}{l|c|c|c|c}{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t-statistic} &amp; \\textbf{P-value}\\\\ \\hline\\text{Intercept} &amp; 81 &amp; 7.9659 &amp; 10.1296 &amp; 0.0000\\\\ \\hline\\text{Inflation rates} &amp; -276 &amp; 233.0748 &amp; -1.1833 &amp; 0.2753\\\\ \\hline\\text{Real interest Rates} &amp; 902 &amp; 279.6949 &amp; 3.2266 &amp; 0.0145\\\\ \\end{array}}$$<\/p>\n<p>Test the statistical significance of the independent variable real rate of interest at the 5% significance level.<\/p>\n<h5>Solution<\/h5>\n<p>We are testing the hypothesis:<\/p>\n<p>\\(H_{0}\\): Real Interest Rate=0 versus \\(H_{a}\\): Real Interest Rate \u2260 0.<\/p>\n<p>The t-statistic is determined as:<\/p>\n<p>$$t=\\frac{\\widehat{b_{j}}-b_{H0}}{S_{\\widehat{b_{j}}}}$$<\/p>\n<p>$$\\begin{align*}t&amp;=\\frac{902-0}{279.6949}\\\\&amp;=3.22\\end{align*}$$<\/p>\n<p>The 5% two-tailed critical t-value with \\(10-2-1 = 7\\) degrees of freedom is 2.3646.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12802 aligncenter\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-t-table-2.36.png\" alt=\"\" width=\"750\" height=\"720\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-t-table-2.36.png 750w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-t-table-2.36-300x288.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-t-table-2.36-400x384.png 400w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/p>\n<p>Since the t-statistic, \\(3.22\\), is greater than the critical value, \\(2.365\\), we reject the null hypothesis and conclude that the real interest rate regression coefficient is statistically significantly different from zero at the 5% significance level.<\/p>\n<h2><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12801\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test.png\" alt=\"\" width=\"1242\" height=\"961\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test.png 1242w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-300x232.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1024x792.png 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-768x594.png 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-400x310.png 400w\" sizes=\"auto, (max-width: 1242px) 100vw, 1242px\" \/><\/h2>\n<blockquote>\n<h2>Question<\/h2>\n<p>Consider the following multiple regression results of the return on capital (ROC) vs. performance measures (profit margin (%), sales and debt ratio).<\/p>\n<p>$$\\small{\\begin{array}{|l|c|}\\hline\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.7906\\\\ \\hline\\text{R Square} &amp; 0.6251\\\\ \\hline\\text{Adjusted R Square} &amp; 0.5715\\\\ \\hline\\text{Standard Error} &amp; 1.1963\\\\ \\hline\\text{Observations} &amp; 25\\\\ \\hline\\end{array}}$$<\/p>\n<p>$$\\small{\\begin{array}{l|c|c|c|c|c}{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t Stat} &amp; \\textbf{P-value} \\\\ \\hline\\text{Intercept} &amp; 8.6531 &amp; 0.9174 &amp; 9.4323 &amp; 0.0000 \\\\ \\hline \\text{Sales} &amp; 0.0009 &amp; 0.0005 &amp; 1.7644 &amp; 0.0922\\\\ \\hline\\text{Debt ratio} &amp; 0.0229 &amp; 0.0165 &amp; 1.3880 &amp; 0.1797 \\\\ \\hline\\text{Profit Margin%} &amp; 0.2996 &amp; 0.0564 &amp; 5.3146 &amp; 0.0000\\\\ \\end{array}}$$<\/p>\n<p>Adil Suleman, CFA, wants to test whether \\(b_{o}\\) is different from zero. Assuming a 5% significance level, the hypothesis used and the result of the hypothesis test are most\u00a0<em>likely:\u00a0<\/em><\/p>\n<p>$$\\begin{array}{|l|l|l|} \\hline<br \/>\n\\text{A} &amp; H_{0}:b_{0}= 0, vs. H_{a}: b_0\u22600 &amp;\u00a0\u00a0{\\text{Fail to reject the null hypothesis and conclude that}\\\\ b_0\u00a0 \\text{ is equal to zero.}}\\\\ \\hline<br \/>\n\\text{B} &amp; H_{0}:b_{0}= 0, vs. H_{a}: b_0=0 &amp;\u00a0\u00a0{\\text{Reject the null hypothesis and conclude that}\\\\ b_0\u00a0 \\text{ is equal to zero.}} \\\\ \\hline<br \/>\n\\text{C} &amp; H_{0}:b_{0}= 0, vs. H_{a}: b_0\u22600 &amp; {\\text{Reject the null hypothesis and conclude that}\\\\ b_0 \\text{ is not equal to zero.}} \\\\ \\hline<br \/>\n\\end{array}$$<\/p>\n<h3>Solution<\/h3>\n<p><strong>The correct answer is C.<\/strong><\/p>\n<p>The hypothesis will be structured as:<\/p>\n<p style=\"text-align: center;\">\\(H_0:b_0\\), v.s. \\(H_a: b_0\u00a0\u22600\\)<\/p>\n<p style=\"text-align: center;\">\\(\\text{t-statistic}= 9.4323\\)<\/p>\n<p>This is a two-tailed test: The critical t-value with 23 degrees of freedom at the 5% significance level is\u00a0\u00b12.069.<\/p>\n<p>Notice that the t-stat is greater than the upper critical value. This prompts the rejection of the null hypothesis and conclusion that \u00a0is significantly different from zero.<\/p>\n<\/blockquote>\n<p>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (c) Formulate a null and an alternative hypothesis about the population value of a regression coefficient, calculate the value of the test statistic, and determine whether to reject the null hypothesis at a given level of significance.<\/em><\/p>\n<p><!-- BOTTOM CTA \u2013 Refined Version --><\/p>\n<div style=\"text-align:center; background-color:#f4f6f9; padding:32px 20px; border-radius:12px; margin-top:40px;\">\n<p>  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:13px 30px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:15.5px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:16px;\n     \"><br \/>\n     Start Free Trial<br \/>\n  <\/a><\/p>\n<p style=\"max-width:680px; margin:0 auto; font-size:15.5px; line-height:1.6; color:#333;\">\n    Strengthen your CFA Level II quantitative skills with exam-style hypothesis testing and statistical inference problems designed to improve model interpretation and exam performance.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hypothesis testing involves testing an assumption regarding a population parameter. A null hypothesis is a condition believed not to be true. We reject the null hypothesis in the presence of enough evidence against it and accept the alternative hypothesis. Hypothesis&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[216,234,230],"class_list":["post-11786","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-hypothesis-testing","tag-quantitative-method","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>Hypothesis Testing II | CFA Level II Notes<\/title>\n<meta name=\"description\" content=\"Learn advanced hypothesis testing concepts, including critical values, significance levels, and interpreting two-tailed t-tests.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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