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CHAPTER 8 MULTIPLE REGRESSION: ESTIMATION AND HYPOTHESIS TESTING QUESTIONS 8.1. ( a ) It measures the change in the mean value of the dependent variable ( Y ) for a unit change in the value of an explanatory variable ( X ), holding the values of all other explanatory variables constant. Mathematically, it is the partial derivative of (mean) Y with respect to the given explanatory variable. ( b ) It measures the proportion, or percentage, of the total variation in the dependent variable, - 2 ) ( Y Y i , explained by all the explanatory variables included in the model. ( c ) Exact linear relationship among the explanatory variables. ( d ) More than one exact linear relationship among the explanatory variables. ( e ) Testing the hypothesis about a single (partial) regression coefficient. ( f ) Testing the hypothesis about two or more partial regression coefficients simultaneously. ( g ) An 2 R value that is adjusted for degrees of freedom. 8.2. ( a ) (1) State the null and alternative hypotheses. (2) Choose the level of significance. (3) Find the t value of the coefficient under the null hypothesis, 0 H . (4) Compare this | t | value with the critical value at the chosen level of significance and the given d.f. (5) If the computed t value exceeds the critical t value, we reject the null hypothesis. Make sure that you use the appropriate one-tailed or two-tailed test. ( b ) Here the null hypothesis is: 0 ... : H 3 2 0 = = = = k B B B 56
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that is, all partial slopes are zero. The alternative hypothesis is that this is not so, that is, one or more partial slope coefficients are nonzero. Here, we use the ANOVA technique and the F test. If the computed F value under the null hypothesis exceeds the critical F value at the chosen level of significance, we reject the null hypothesis. Otherwise, we do not reject it. Make sure that the numerator and denominator d.f. are properly counted. Note : In both ( a ) and ( b ), instead of choosing the level of significance in advance, obtain the p value of the estimated test statistic. If it is reasonably low, you can reject the null hypothesis. 8.3. ( a ) True . This is obvious from the formula relating the two 2 R s. ( b ) False . Use the F test. ( c ) False . When 2 R = 1, the value of F is infinite. But when it is zero, the F value is also zero. ( d ) True , which can be seen from the normal and t distribution tables. ( e ) True . It can be shown that 32 3 2 12 ) ( b B B b E + = , where 32 b is the slope coefficient in the regression of 3 X on 2 X . From this relationship, the conclusion follows. ( f ) False . It is statistically different from zero, not 1. ( g ) False . We also need to know the level of significance. ( h ) False . By the overall significance we mean that all partial regression coefficients are not simultaneously equal to zero, or that 2 R is different from zero. ( i ) Partially true . If our concern is only with a single regression coefficient, then we use the t test in both cases. But if we are interested in testing the joint significance of two or more partial regression coefficients, the t test will not do; we will have to use the F test.
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