chapter6_s - CHAPTER 6 SOLUTIONS TO PROBLEMS 6.1 The...

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CHAPTER 6 SOLUTIONS TO PROBLEMS 6.1 The generality is not necessary. The t statistic on roe 2 is only about .30, which shows that roe 2 is very statistically insignificant. Plus, having the squared term has only a minor effect on the slope even for large values of roe . (The approximate slope is .0215 .00016 roe , and even when roe = 25 – about one standard deviation above the average roe in the sample – the slope is .211, as compared with .215 at roe = 0.) 6.3 (i) The turnaround point is given by 1 ˆ β /(2| 2 ˆ |), or .0003/(.000000014) 21,428.57; remember, this is sales in millions of dollars. (ii) Probably. Its t statistic is about –1.89, which is significant against the one-sided alternative H 0 : 1 < 0 at the 5% level ( cv –1.70 with df = 29). In fact, the p -value is about .036. (iii) Because sales gets divided by 1,000 to obtain salesbil , the corresponding coefficient gets multiplied by 1,000: (1,000)(.00030) = .30. The standard error gets multiplied by the same factor. As stated in the hint, salesbil 2 = sales /1,000,000, and so the coefficient on the quadratic gets multiplied by one million: (1,000,000)(.0000000070) = .0070; its standard error also gets multiplied by one million. Nothing happens to the intercept (because rdintens has not been rescaled) or to the R 2 : n rdintens = 2.613 + .30 salesbil – .0070 salesbil 2 (0.429) (.14) (.0037) n = 32, R 2 = .1484. (iv) The equation in part (iii) is easier to read because it contains fewer zeros to the right of the decimal. Of course the interpretation of the two equations is identical once the different scales are accounted for. 6.5 This would make little sense. Performances on math and science exams are measures of outputs of the educational process, and we would like to know how various educational inputs and school characteristics affect math and science scores. For example, if the staff-to-pupil ratio has an effect on both exam scores, why would we want to hold performance on the science test fixed while studying the effects of staff on the math pass rate? This would be an example of controlling for too many factors in a regression equation. The variable scill could be a dependent variable in an identical regression equation. 6.7 The second equation is clearly preferred, as its adjusted R -squared is notably larger than that in the other two equations. The second equation contains the same number of estimated 27
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parameters as the first, and the one fewer than the third. The second equation is also easier to interpret than the third. 6.9 (i) Because ˆ exp( 1.96 ) 1 σ −< 2 ˆ / 2) exp(1.96 and , the point prediction is always above the lower bound. The only issue is whether the point prediction is below the upper bound. This is the case when 2 ˆ exp( / 2) 1 > ˆ exp( ) or, taking logs, 2 ˆˆ / 2 1.96 , or ˆ 2(1.96) 3.92 = ˆ 3.92 . Therefore, the point prediction is in the approximate 95% prediction interval for . Because ˆ is the estimated standard deviation in the regression with log( y ) as the dependent variable, 3.92 is a very large value for the estimated standard deviation of the error, which is on the order of 400 percent. Most of the time, the estimated SER is well below that.
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chapter6_s - CHAPTER 6 SOLUTIONS TO PROBLEMS 6.1 The...

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