{"id":36740,"date":"2021-10-29T17:17:20","date_gmt":"2021-10-29T17:17:20","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=36740"},"modified":"2026-04-20T08:37:18","modified_gmt":"2026-04-20T08:37:18","slug":"hypothesis-testing-in-regression-analysis","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/hypothesis-testing-in-regression-analysis\/","title":{"rendered":"Hypothesis Testing in Regression Analysis"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Introduction to Linear Regression (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 7)\",\n  \"description\": \"This CFA\u00ae Level I Quantitative Methods lesson introduces linear regression and its role in financial analysis. The video explains the structure of a simple linear regression model, including dependent and independent variables, and shows how the least squares criterion is used to estimate and interpret regression coefficients. It covers key assumptions of the model, residual analysis, goodness-of-fit measures such as R-squared and the F-statistic, and the use of ANOVA. The lesson also walks through hypothesis testing for regression coefficients, prediction and prediction intervals, and different functional forms of simple linear regression, with a strong emphasis on exam-relevant interpretation.\",\n  \"uploadDate\": \"2021-11-04\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/6ouI_27iDVM\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=6ouI_27iDVM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/6ouI_27iDVM\",\n  \"duration\": \"PT56M38S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the correct t-statistic for the slope and is it statistically significant at the 95% confidence level?\",\n    \"text\": \"Neeth Shinu, CFA, is forecasting the price elasticity of supply for a certain product. Shinu uses the quantity of the product supplied for the past 5 months as the dependent variable and the price per unit of the product as the independent variable. The regression results show a slope coefficient of 0.26 with a standard error of 0.012.\\n\\nWhich of the following most likely reports the correct value of the t-statistic for the slope and most accurately evaluates its statistical significance with 95% confidence?\\n\\nA. t = 21.67; slope is significantly different from zero.\\nB. t = 3.18; slope is significantly different from zero.\\nC. t = 22.57; slope is not significantly different from zero.\",\n    \"answerCount\": 3,\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. t = 21.67; slope is significantly different from zero.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. t = 3.18; slope is significantly different from zero.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. t = 22.57; slope is not significantly different from zero.\"\n      }\n    ],\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. t = 21.67; slope is significantly different from zero.\",\n      \"commentary\": \"The t-statistic for the slope coefficient is calculated as t = b\u0302\u2081 \/ SE(b\u0302\u2081) = 0.26 \/ 0.012 = 21.67. With 5 observations, the degrees of freedom are n \u2212 2 = 3. The two-tailed critical t-value at the 95% confidence level is approximately 3.18. Since |21.67| > 3.18, the null hypothesis is rejected and the slope coefficient is statistically different from zero.\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/hypothesis-testing-in-regression-analysis\/\"\n    },\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/6ouI_27iDVM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Hypothesis testing is used to confirm if the estimated regression coefficients bear any statistical significance. &nbsp;Either the confidence interval approach or the t-test approach can be used in hypothesis testing. In this section, we will explore the t-test approach.<\/p>\n<h2><strong>The t-test Approach<\/strong><\/h2>\n<p>The following are the steps followed in the performance of the t-test:<\/p>\n<ol>\n<li>Set the significance level for the test.<\/li>\n<li>Formulate the null and the alternative hypotheses.<\/li>\n<li>Calculate the t-statistic using the formula below:\n<p>$$t=\\frac{\\widehat{b_1}-b_1}{s_{\\widehat{b_1}}}$$<\/p>\n<p>Where:<\/p>\n<p>\\(b_1\\) = True slope coefficient.<\/p>\n<p>\\(\\widehat{b_1}\\) = Point estimate for \\(b_1\\)<\/p>\n<p>\\(b_1 s_{\\widehat{b_1\\ }}\\) = Standard error of the regression coefficient.<\/p>\n<\/li>\n<li>Compare the absolute value of the t-statistic to the critical t-value (t_c). Reject the null hypothesis if the absolute value of the t-statistic is greater than the critical t-value i.e., \\(t\\ &gt;\\ +\\ t_{critical}\\ or\\ t\\ &lt;\\ \u2013t_{\\text{critical}}\\).<\/li>\n<\/ol>\n<div style=\"background-color:#f5f7fa; padding:20px; text-align:center; margin:30px 0; border-radius:8px;\">\n  <div style=\"max-width:620px; margin:0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display:flex; align-items:center; justify-content:center; width:100%; padding:10px 20px; border:2px solid #1a73e8; border-radius:999px; text-decoration:none; color:#1a73e8; font-size:15px; font-weight:600; line-height:1.4;\">\n      Test regression significance with exam-focused practice in our Free Trial\n    <\/a>\n  <\/div>\n<\/div>\n<h4><strong>Example: Hypothesis Testing of the Significance of Regression Coefficients<\/strong><\/h4>\n<p>An analyst generates the following output from the regression analysis of inflation on unemployment:<\/p>\n<p>$$\\small{\\begin{array}{llll}\\hline{}&amp; \\textbf{Regression Statistics} &amp;{}&amp;{}\\\\ \\hline{}&amp; \\text{Multiple R} &amp; 0.8766 &amp;{} \\\\ {}&amp; \\text{R Square} &amp; 0.7684 &amp;{} \\\\ {}&amp; \\text{Adjusted R Square} &amp; 0.7394 &amp; {}\\\\ {}&amp; \\text{Standard Error} &amp; 0.0063 &amp;{}\\\\ {}&amp; \\text{Observations} &amp; 10 &amp;{}\\\\ \\hline {}&amp; &amp; &amp; \\\\ \\hline{} &amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t-Stat}\\\\ \\hline \\text{Intercept} &amp; 0.0710 &amp; 0.0094 &amp; 7.5160 \\\\\\text{Forecast (Slope)} &amp; -0.9041 &amp; 0.1755 &amp; -5.1516\\\\ \\hline\\end{array}}$$<\/p>\n<p>At the 5% significant level, test the null hypothesis that the slope coefficient is significantly different from one, that is,<\/p>\n<p>$$ H_{0}: b_{1} = 1\\ vs. \\ H_{a}: b_{1}\u22601 $$<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The calculated t-statistic, \\(\\text{t}=\\frac{\\widehat{b_{1}}-b_1}{\\widehat{S_{b_{1}}}}\\) is equal to:<\/p>\n<p>$$\\begin{align*}\\text{t}&amp; = \\frac{-0.9041-1}{0.1755}\\\\&amp; = -10.85\\end{align*}$$<\/p>\n<p>The critical two-tail t-values from the table with \\(n-2=8\\) degrees of freedom are:<\/p>\n<p>$$\\text{t}_{c}=\u00b12.306$$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12774 aligncenter\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/02\/cfa-level-2-t-table-2.3.png\" alt=\"\" width=\"735\" height=\"720\"><\/p>\n<p>Notice that \\(|t|&gt;t_{c}\\) i.e., (\\(10.85&gt;2.306\\))<\/p>\n<p>Therefore, we reject the null hypothesis and conclude that the estimated slope coefficient is statistically different from one.<\/p>\n<p>Note that we used the confidence interval approach and arrived at the same conclusion.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Neeth Shinu, CFA, is forecasting price elasticity of supply for a certain product. Shinu uses the quantity of the product supplied for the past 5months as the dependent variable and the price per unit of the product as the independent variable. The regression results are shown below.<\/p>\n<p>$$\\small{\\begin{array}{lccccc}\\hline \\textbf{Regression Statistics} &amp; &amp; &amp; &amp; &amp; \\\\ \\hline \\text{Multiple R} &amp; 0.9971 &amp; {}&amp; {}&amp;{}\\\\ \\text{R Square} &amp; 0.9941 &amp; &amp; &amp; \\\\ \\text{Adjusted R Square} &amp; 0.9922 &amp; &amp; &amp; &amp; \\\\ \\text{Standard Error} &amp; 3.6515 &amp; &amp; &amp; \\\\ \\text{Observations} &amp; 5 &amp; &amp; &amp; \\\\ \\hline {}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t Stat} &amp; \\textbf{P-value}\\\\ \\hline\\text{Intercept} &amp; -159 &amp; 10.520 &amp; (15.114) &amp; 0.001\\\\ \\text{Slope} &amp; 0.26 &amp; 0.012 &amp; 22.517 &amp; 0.000\\\\ \\hline\\end{array}}$$<\/p>\n<p>Which of the following <em>most likely<\/em> reports the correct value of the t-statistic for the slope and <em>most accurately<\/em> evaluates its statistical significance with 95% confidence?<\/p>\n<p>&nbsp; &nbsp; A. \\(t=21.67\\);&nbsp;slope is significantly different from zero.<\/p>\n<p>&nbsp; &nbsp; B. \\(t= 3.18\\); slope is significantly different from zero.<\/p>\n<p>&nbsp; &nbsp; C. \\(t=22.57\\);&nbsp;slope is not significantly different from zero.<\/p>\n<h3>Solution<\/h3>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>The t-statistic is calculated using the formula:<\/p>\n<p>$$\\text{t}=\\frac{\\widehat{b_{1}}-b_1}{\\widehat{S_{b_{1}}}}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li>\\(b_{1}\\) = True slope coefficient<\/li>\n<li>\\(\\widehat{b_{1}}\\) = Point estimator for \\(b_{1}\\)<\/li>\n<li>\\(\\widehat{S_{b_{1}}}\\) = Standard error of the regression coefficient<\/li>\n<\/ul>\n<p>$$\\begin{align*}\\text{t}&amp;=\\frac{0.26-0}{0.012}\\\\&amp;=21.67\\end{align*}$$<\/p>\n<p>The critical two-tail t-values from the t-table with \\(n-2 = 3\\) degrees of freedom are:<\/p>\n<p>$$t_{c}=\u00b13.18$$<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12775\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/02\/cfa-level-2-t-table-3.18.png\" alt=\"\" width=\"735\" height=\"720\"><\/p>\n<p style=\"text-align: left;\">Notice that \\(|t|&gt;t_{c}\\) (i.e \\(21.67&gt;3.18\\)).<\/p>\n<p>Therefore, the null hypothesis can be rejected. Further, we can conclude that the estimated slope coefficient is statistically different from zero.<\/p>\n<\/blockquote>\n<div style=\"text-align:center; margin:40px 0;\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" style=\"background:#1a73e8; color:#ffffff; padding:14px 26px; border-radius:999px; text-decoration:none; font-weight:700; font-size:16px; display:inline-block;\">\nStart Free Trial \u2192\n<\/a>\n<p style=\"margin-top:10px; font-size:14px; color:#555;\">\nAccess CFA Level I practice questions, video lessons, and study notes covering regression analysis, hypothesis testing, and statistical inference in quantitative methods.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hypothesis testing is used to confirm if the estimated regression coefficients bear any statistical significance. &nbsp;Either the confidence interval approach or the t-test approach can be used in hypothesis testing. In this section, we will explore the t-test approach. The&#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-36740","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>Hypothesis Testing in Regression | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore hypothesis testing in regression analysis, including t-tests, p-values, and their role in evaluating multiple regression models. 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