{"id":11804,"date":"2021-03-01T19:21:11","date_gmt":"2021-03-01T19:21:11","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11804"},"modified":"2026-07-07T20:07:25","modified_gmt":"2026-07-07T20:07:25","slug":"interpreting-results-hypothesis-tests-regression-coefficients","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/interpreting-results-hypothesis-tests-regression-coefficients\/","title":{"rendered":"Interpreting the Results of Hypothesis Tests of Regression Coefficients"},"content":{"rendered":"<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-1.181.png\",\n  \"caption\": \"Rejection Regions - Two-Sided 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><\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8?si=Q-8gbHd1YwdPuDCt\" 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><\/p>\n<p>The following results are obtained from regressing the price of the US Dollar Index (USDX) on inflation rates and real interest rates.<\/p>\n<p>$$\\small{\\begin{array}{l|c}\\textbf{Regression Statistics}\\\\ \\hline\\text{Multiple R} &amp; 0.8264\\\\ \\hline\\text{R Square} &amp; 0.6830\\\\ \\hline\\text{Adjusted R Square} &amp; 0.5924\\\\ \\hline\\text{Standard Error} &amp; 5.3537\\\\ \\hline\\text{Observations} &amp; 10\\\\ \\end{array}}$$<\/p>\n<p>$$\\small{\\begin{array}{l|l|l|l|l}{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\textbf{t Stat} &amp; \\textbf{P-value}\\\\ \\hline\\text{Intercept} &amp; 81 &amp; 7.9659 &amp; 10.1296 &amp; 0.0000\\\\ \\hline\\text{Inflation rates} &amp; -276 &amp; 233.0748 &amp; -1.1833 &amp; 0.2753\\\\ \\hline\\text{Real interest Rates} &amp; 902 &amp; 279.6949 &amp; 3.2266 &amp; 0.0145\\\\ \\end{array}}$$<\/p>\n<p>Test the null hypothesis that the inflation rate is equal to 0 against the alternative hypothesis that it is not equal to 0 at the 5% significance level and interpret the results<\/p>\n<div style=\"margin: 18px 0;\"><a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff ; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8 ;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">Master CFA Level 2 multiple regression equations with AnalystPrep\u2019s Free Trial. <\/a><\/div>\n<p>We are testing the hypothesis:<\/p>\n<p style=\"text-align: center;\">\\(H_{o}:b_1=0\\) vs. \\(H_1:b_1\u22600\\)<\/p>\n<p>The 5% two-tailed critical t-value with \\(10 &#8211; 2 &#8211; 1 = 7\\) degrees of freedom is 2.365<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12802\" 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>The null hypothesis is rejected if the t-statistic is either greater than 2.365 or smaller than \u20132.365 (the critical value).<\/p>\n<p>$$t=\\frac{\\widehat{b_{j}}-b_{H0}}{S_{\\widehat{b_{j}}}}$$<\/p>\n<p>$$\\begin{align*}t&amp;=\\frac{-276-0}{233.0748}&amp;\\\\&amp;=-1.18\\end{align*}$$<\/p>\n<p>The <em>t<\/em>-statistic of -1.18 is between the upper and lower critical <em>t<\/em>-values of \u20132.3646 and 2.3646. We cannot reject the null hypothesis and must conclude that the inflation rate regression coefficient is NOT statistically significantly different from 0 at the 5% significance level.<\/p>\n<p><em><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12806\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181.png\" alt=\"\" width=\"1242\" height=\"961\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181.png 1242w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181-300x232.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181-1024x792.png 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181-768x594.png 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-two-sided-test-1.181-400x310.png 400w\" sizes=\"auto, (max-width: 1242px) 100vw, 1242px\" \/><\/em>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (d) Interpret the results of hypothesis tests of regression coefficients<\/em><\/p>\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a> <\/p>\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Strengthen your understanding of multiple regression equations in CFA Level 2 Quantitative Methods with study notes, practice questions, and mock exams.\n<\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>The following results are obtained from regressing the price of the US Dollar Index (USDX) on inflation rates and real interest rates. $$\\small{\\begin{array}{l|c}\\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;&#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":[216,241,230],"class_list":["post-11804","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-interpreting-the-results-of-hypothesis-tests-of-regression-coefficients","tag-quantitative-method","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Hypothesis Tests of Regression Coefficients | CFA L2<\/title>\n<meta name=\"description\" content=\"Learn how to interpret hypothesis tests for regression coefficients using t-tests, p-values, and confidence intervals. 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