{"id":1420,"date":"2019-05-22T11:23:00","date_gmt":"2019-05-22T11:23:00","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=1420"},"modified":"2026-01-19T16:19:12","modified_gmt":"2026-01-19T16:19:12","slug":"hypothesis-tests-and-confidence-intervals-in-multiple-regression","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/frm\/part-1\/quantitative-analysis\/hypothesis-tests-and-confidence-intervals-in-multiple-regression\/","title":{"rendered":"Hypothesis Tests and Confidence Intervals in Multiple Regression"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/multiple-regression-slide-1\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/slide-1.png\",\n  \"url\": 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\"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/hypothesis-formulation-multiple-regression\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-formulation-in-multiple-regression.png\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-formulation-in-multiple-regression.png\",\n  \"caption\": \"Hypothesis Formulation in Multiple Regression\",\n  \"width\": 303,\n  \"height\": 91,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/frm\/part-1\/quantitative-analysis\/hypothesis-tests-and-confidence-intervals-in-multiple-regression\/\"\n  }\n}\n<\/script><\/p>\n<p><script 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}\n}\n<\/script><\/p>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/r-squared-formula\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula.png\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula.png\",\n  \"caption\": \"R-Squared Formula Illustration\",\n  \"width\": 539,\n  \"height\": 105,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/frm\/part-1\/quantitative-analysis\/hypothesis-tests-and-confidence-intervals-in-multiple-regression\/\"\n  }\n}\n<\/script><\/p>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/t-table-multiple-regression\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/2.9t-table.png\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/2.9t-table.png\",\n  \"caption\": \"t-Table (2.9) for Multiple Regression\",\n  \"width\": 352,\n  \"height\": 108,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/frm\/part-1\/quantitative-analysis\/hypothesis-tests-and-confidence-intervals-in-multiple-regression\/\"\n  }\n}\n<\/script><\/p>\n<h3 id=\"mce_22\" class=\"editor-rich-text__tinymce mce-content-body\" contenteditable=\"true\" data-is-placeholder-visible=\"false\">[vsw id=&#8221;7MaeKHnLjfk&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;]<\/h3>\n<p>After completing this reading you should be able to:<\/p>\n<ul>\n<li>Construct, apply, and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression.<\/li>\n<li>Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a multiple regression.<\/li>\n<li>Interpret the \\(F\\)-statistic.<\/li>\n<li>Interpret tests of a single restriction involving multiple coefficients.<\/li>\n<li>Interpret confidence sets for multiple coefficients.<\/li>\n<li>Identify examples of omitted variable bias in multiple regressions.<\/li>\n<li>Interpret the \\({ R }^{ 2 }\\) and adjusted \\({ R }^{ 2 }\\) in a multiple regression.<\/li>\n<\/ul>\n<h2>Hypothesis Tests and Confidence Intervals for a Single Coefficient<\/h2>\n<p>This section is about the calculation of the standard error, hypotheses testing, and confidence interval construction for a single regression in a multiple regression equation.<\/p>\n<h3>Introduction<\/h3>\n<p>In a previous chapter, we looked at simple linear regression where we deal with just one regressor (independent variable). The response (dependent variable) is assumed to be affected by just one independent variable.\u00a0<br \/>\n <strong>M<\/strong><b>ultiple regression, <\/b>on the other hand<b>,<\/b>\u00a0simultaneously considers the influence of multiple explanatory variables on a response variable Y. We may want to establish the confidence interval of one of the independent variables. We may want to evaluate whether any particular independent variable has a significant effect on the dependent variable. Finally, We may also want to establish whether the independent variables as a group have a significant effect on the dependent variable. In this chapter, we delve into ways all this can be achieved.<\/p>\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2730 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/slide-1.png\" alt=\"\" width=\"555\" height=\"313\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/slide-1.png 555w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/slide-1-300x169.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/slide-1-400x226.png 400w\" sizes=\"auto, (max-width: 555px) 100vw, 555px\" \/><\/h3>\n<h3>Hypothesis Tests for a single coefficient<\/h3>\n<p>Suppose that we are testing the hypothesis that the true coefficient \\({ \\beta }_{ j }\\) on the \\(j\\)th regressor takes on some specific value \\({ \\beta }_{ j,0 }\\). Let the alternative hypothesis be two-sided. Therefore, the following is the mathematical expression of the two hypotheses:<\/p>\n<p>$$ { H }_{ 0 }:{ \\beta }_{ j }={ \\beta }_{ j,0 }\\quad vs.\\quad { H }_{ 1 }:{ \\beta }_{ j }\\neq { \\beta }_{ j,0 } $$<\/p>\n<p>This expression represents the two-sided alternative. The following are the steps to follow while testing the null hypothesis:<\/p>\n<ol type=\"1\">\n<li>Computing the coefficient\u2019s standard error.<\/li>\n<li>Computing the \\(t\\)-statistic, as previously described:<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2733 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-test-under-multiple-regression.png\" alt=\"\" width=\"534\" height=\"191\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-test-under-multiple-regression.png 534w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-test-under-multiple-regression-300x107.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-test-under-multiple-regression-400x143.png 400w\" sizes=\"auto, (max-width: 534px) 100vw, 534px\" \/><\/li>\n<li>Computing the test\u2019s \\(p-value\\) as previously described:\n<p>$$ p-value=2\\Phi \\left( -|{ t }^{ act }| \\right) $$<\/p>\n<\/li>\n<li>Also, the \\(t\\)-statistic can be compared to the critical value corresponding to the significance level that is desired for the test.<\/li>\n<\/ol>\n<h3>\u00a0<\/h3>\n<h3>Confidence Intervals for a Single Coefficient<\/h3>\n<p>The confidence interval for a regression coefficient in multiple regression is calculated and interpreted the same way as it is in simple linear regression.\u00a0<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2740 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/CI.png\" alt=\"\" width=\"526\" height=\"193\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/CI.png 526w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/CI-300x110.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/CI-400x147.png 400w\" sizes=\"auto, (max-width: 526px) 100vw, 526px\" \/><\/p>\n<p>The t-statistic has <u><b>n \u2013 k \u2013 1 <\/b><\/u>degrees of freedom where k = number of independents<\/p>\n<p>Supposing that an interval contains the true value of \\({ \\beta }_{ j }\\) with a probability of 95%. This is simply the 95% two-sided confidence interval for \\({ \\beta }_{ j }\\). The implication here is that the true value of \\({ \\beta }_{ j }\\) is contained in 95% of all possible randomly drawn variables.<\/p>\n<p>Alternatively, the 95% two-sided confidence interval for \\({ \\beta }_{ j }\\) is the set of values that are impossible to reject when a two-sided hypothesis test of 5% is applied. Therefore, with a large sample size:<\/p>\n<p>$$ 95\\%\\quad confidence\\quad interval\\quad for\\quad { \\beta }_{ j }=\\left[ { \\hat { \\beta } }_{ j }-1.96SE\\left( { \\hat { \\beta } }_{ j } \\right) ,{ \\hat { \\beta } }_{ j }+1.96SE\\left( { \\hat { \\beta } }_{ j } \\right) \\right] $$<\/p>\n<h2>Tests of Joint Hypotheses<\/h2>\n<p>In this section, we consider the formulation of the joint hypotheses on multiple regression coefficients. We will further study the application of an \\(F\\)-statistic in their testing.<\/p>\n<h3>Hypotheses Testing on Two or More Coefficients<\/h3>\n<h4>Joint Null Hypothesis<\/h4>\n<p>In multiple regression, we <u><b>canno<\/b><\/u><u><b>t<\/b><\/u> test the null hypothesis that <i>all <\/i>slope coefficients are equal 0 based on <i>t<\/i>-tests that <i>each individual <\/i>slope coefficient equals 0.\u00a0Why? individual t-tests do not account for the effects of <u><b>interactions<\/b><\/u> among the independent variables.<\/p>\n<p>For this reason, we conduct the <u><b>F-test <\/b><strong>which uses the F-statistic<\/strong><b>.\u00a0<\/b><\/u>The F-test tests the null hypothesis that all of the slope coefficients in the multiple regression model are jointly equal to 0, .i.e.,<\/p>\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2743 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-formulation-in-multiple-regression.png\" alt=\"\" width=\"303\" height=\"91\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-formulation-in-multiple-regression.png 303w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/hypothesis-formulation-in-multiple-regression-300x91.png 300w\" sizes=\"auto, (max-width: 303px) 100vw, 303px\" \/><\/h4>\n<h4>\\(F\\)-Statistic<\/h4>\n<p>The F-statistic, <u><b><i>which is always a one-tailed test<\/i><\/b><\/u><i>, <\/i>is calculated as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2746 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/F-test.png\" alt=\"\" width=\"548\" height=\"283\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/F-test.png 548w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/F-test-300x155.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/F-test-400x207.png 400w\" sizes=\"auto, (max-width: 548px) 100vw, 548px\" \/><\/p>\n<h4>\u00a0<\/h4>\n<h4>\u00a0<\/h4>\n<p>To determine whether at least one of the coefficients is statistically significant, the calculated F-statistic is compared with the one-tailed critical F-value, at the appropriate level of significance.<\/p>\n<p>Decision rule:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2748 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/decision-rule.png\" alt=\"\" width=\"407\" height=\"52\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/decision-rule.png 407w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/decision-rule-300x38.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/decision-rule-400x51.png 400w\" sizes=\"auto, (max-width: 407px) 100vw, 407px\" \/><\/p>\n<p>Rejection of the null hypothesis at a stated level of significance indicates that at least one of the coefficients is significantly different than zero, i.e, at least one of the independent variables in the regression model makes a significant contribution to the dependent variable.<\/p>\n<h4><em>Example<\/em><\/h4>\n<p>An analyst runs a regression of monthly value-stock returns on four independent variables over 48 months.<\/p>\n<p>The total sum of squares for the regression is 360, and the sum of squared errors is 120.<\/p>\n<p>Test the null hypothesis at the 5% significance level (95% confidence) that all the four independent variables are equal to zero.<\/p>\n<h4><em><strong>Solution<\/strong><\/em><\/h4>\n<p>\\({ H }_{ 0 }:{ \\beta }_{ 1 }=0,{ \\beta }_{ 2 }=0,\\dots ,{ \\beta }_{\u00a04 }=0 \\)<\/p>\n<p>Versus<\/p>\n<p>\\({ H }_{ 1 }:{ \\beta }_{ j }\\neq 0\\) (at least one j is not equal to zero, j=1,2&#8230; k )<\/p>\n<p>ESS = TSS \u2013 SSR = 360 \u2013 120 = 240<\/p>\n<p>The calculated test statistic = (ESS\/k)\/(SSR\/(n-k-1))<\/p>\n<p>=(240\/4)\/(120\/43) = 21.5<\/p>\n<p>\\({ F }_{ 43 }^{ 4 }\\) is approximately 2.44 at 5% significance level.<\/p>\n<p>Decision: Reject H<sub>0<\/sub>.<\/p>\n<p>Conclusion: <b>at <\/b><b>least one <\/b>of the 4 independents is significantly different than zero.<\/p>\n<h4>\u00a0<\/h4>\n<h2>Omitted Variable Bias in Multiple Regression<\/h2>\n<p>This is the bias in the OLS estimator arising when at least one included regressor gets collaborated with an omitted variable. The following conditions must be satisfied for an omitted variable bias to occur:<\/p>\n<ul>\n<li>There must be a correlation between at least one of the included regressors and the omitted variable.<\/li>\n<li>The dependent variable \\(Y\\) must be determined by the omitted variable.<\/li>\n<\/ul>\n<h3>\u00a0<\/h3>\n<h2>Practical Interpretation of the \\({ R }^{ 2 }\\) and the adjusted \\({ R }^{ 2 }\\), \\({ \\bar { R } }^{ 2 }\\)<\/h2>\n<p>To determine the accuracy within which the OLS regression line fits the data, we apply the coefficient of determination\u00a0and the <b>regression\u2019s standard error<\/b>.\u00a0<\/p>\n<p>The <b>coefficient of determination, <\/b>represented by \\({ R }^{ 2 }\\), is a measure of the \u201cgoodness of fit\u201d of the regression. It is interpreted as the percentage of variation in the dependent variable explained by the independent variables<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2756 alignnone\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula.png\" alt=\"\" width=\"539\" height=\"105\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula.png 539w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula-300x58.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/r-squared-formula-400x78.png 400w\" sizes=\"auto, (max-width: 539px) 100vw, 539px\" \/><\/p>\n<p>\\({ R }^{ 2 }\\) is not a reliable indicator of the explanatory power of a multiple regression model.Why?\u00a0\\({ R }^{ 2 }\\) <u><b>almost always <\/b><\/u>increases as new independent variables are added to the model, even if the marginal contribution of the new variable is not statistically significant. Thus, a high \\({ R }^{ 2 }\\) may reflect the impact of a large set of independents rather than how well the set explains the dependent.This problem is solved by the use of the adjusted \\({ R }^{ 2 }\\) (extensively covered in chapter 8)<\/p>\n<p>The following are the factors to watch out when guarding against applying the \\({ R }^{ 2 }\\) or the \\({ \\bar { R } }^{ 2 }\\):<\/p>\n<ol type=\"a\">\n<li>An added variable doesn\u2019t have to be statistically significant just because the \\({ R }^{ 2 }\\) or the \\({ \\bar { R } }^{ 2 }\\) has increased.<\/li>\n<li>It is not always true that the regressors are a true cause of the dependent variable, just because there is a high \\({ R }^{ 2 }\\) or \\({ \\bar { R } }^{ 2 }\\).<\/li>\n<li>It is not necessary that there is no omitted variable bias just because we have a high \\({ R }^{ 2 }\\) or \\({ \\bar { R } }^{ 2 }\\).<\/li>\n<li>It is not necessarily true that we have the most appropriate set of regressors just because we have a high \\({ R }^{ 2 }\\) or \\({ \\bar { R } }^{ 2 }\\).<\/li>\n<li>It is not necessarily true that we have an inappropriate set of regressors just because we have a low \\({ R }^{ 2 }\\) or \\({ \\bar { R } }^{ 2 }\\).<\/li>\n<\/ol>\n<p>Question 1<\/p>\n<p>An economist tests the hypothesis that GDP growth in a certain country can be explained by interest rates and inflation.<\/p>\n<p>Using some 30 observations, the analyst formulates the following regression equation:<\/p>\n<p>$$ GDP growth = { \\hat { \\beta } }_{\u00a00 } + { \\hat { \\beta } }_{ 1 } Interest+ { \\hat { \\beta } }_{\u00a02 }\u00a0Inflation $$<\/p>\n<p>Regression estimates are as follows:<\/p>\n<table width=\"616\">\n<tbody>\n<tr>\n<td width=\"205\">\u00a0<\/td>\n<td width=\"205\">\n<p><b>Coefficient<\/b><\/p>\n<\/td>\n<td width=\"205\">\n<p><b>Standard error<\/b><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"205\">\n<p>Intercept<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.10<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.5%<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"205\">\n<p>Interest rates<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.20<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.05<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"205\">\n<p>Inflation<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.15<\/p>\n<\/td>\n<td width=\"205\">\n<p>0.03<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Is the coefficient for interest rates significant at 5%?<\/p>\n<ol type=\"A\">\n<li>Since the test statistic\u00a0&lt; t-critical, we accept H<sub>0<\/sub>; the interest rate coefficient is\u00a0<em><strong>not<\/strong>\u00a0<\/em>significant at the 5% level.<\/li>\n<li>Since the test statistic &gt;\u00a0t-critical, we reject H<sub>0<\/sub>; the interest rate coefficient is <em><strong>not<\/strong> <\/em>significant at the 5% level.<\/li>\n<li>Since the test statistic &gt; t-critical, we reject H<sub>0<\/sub>; the interest rate coefficient is significant at the 5% level.<\/li>\n<li>Since the test statistic\u00a0&lt;\u00a0t-critical, we accept H<sub>1<\/sub>; the interest rate coefficient is<b><i> <\/i><\/b>significant at the 5% level.<\/li>\n<\/ol>\n<p>The correct answer is\u00a0<strong>C<\/strong>.<\/p>\n<p>We have GDP growth = 0.10 + 0.20(Int) + 0.15(Inf)<\/p>\n<p>Hypothesis:<\/p>\n<p>$$ { H\u00a0}_{ 0 }:{ \\hat { \\beta } }_{ 1 } = 0 \\quad vs \\quad { H\u00a0}_{\u00a01 }:{ \\hat { \\beta } }_{ 1 }\u22600 $$<\/p>\n<p>The test statistic is:<\/p>\n<p>$$ t = \\left( \\frac {\u00a00.20 &#8211; 0 }{\u00a00.05 } \\right)\u00a0 = 4 $$<\/p>\n<p>The critical value is t<sub>(\u03b1\/2, n-k-1)<\/sub> = t<sub>0.025,27\u00a0<\/sub>= 2.052 (which can be found on the t-table).<\/p>\n<p><b><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2717\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/2.9t-table.png\" alt=\"t-table-25-29\" width=\"352\" height=\"108\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/2.9t-table.png 352w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2018\/09\/2.9t-table-300x92.png 300w\" sizes=\"auto, (max-width: 352px) 100vw, 352px\" \/>Decision<\/b>: Since test statistic &gt; t-critical, we reject H<sub>0<\/sub>.<\/p>\n<p><b>Conclusion<\/b>: The interest rate coefficient is significant at the 5% level.<\/p>\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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\n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-1\">\n            <\/div>\n            \n","protected":false},"excerpt":{"rendered":"<p>[vsw id=&#8221;7MaeKHnLjfk&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;] After completing this reading you should be able to: Construct, apply, and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression. Construct, apply, and interpret joint hypothesis tests and&#8230;<\/p>\n","protected":false},"author":3,"featured_media":1521,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7,16],"tags":[],"class_list":["post-1420","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-part-1","category-quantitative-analysis","blog-post","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Confidence Intervals in Multiple Regression | AnalystPrep - FRM Part 1<\/title>\n<meta name=\"description\" content=\"After completing this reading you should be able to construct, apply, and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/frm\/part-1\/quantitative-analysis\/hypothesis-tests-and-confidence-intervals-in-multiple-regression\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Confidence Intervals in Multiple Regression | AnalystPrep - 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