1. Define the following terms (1 point each): alternative hypothesis
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1 1 1. Define the following terms (1 point each): alternative hypothesis One of three hypotheses indicating that the parameter is not zero; one states the parameter is not equal to zero, one states the parameter is larger than zero, and one states the parameter is smaller than zero. proaility value An area under the sampling distriution of t under the assumption that the null hypothesis is true: the area eyond calculated t for a two-tailed test; the area aove calculated t for an upper-tailed test; and the area elow calculated t for a lower-tailed test. Type I error rate The proaility of rejecting the null hypothesis when it is true.
2 . Using the data construct a scatter plot with GPA as the dependent variale GPA AT Plot of GPA Versus AT
3 3 3. For these data, would it e appropriate to report Y X No. Calculating Y X? If not, why not is ased on the assumption of equal conditional variances. The plot suggests this assumption is violated. There is relatively little conditional variance when AT is low (e.g., 1000) and relatively large conditional variance when AT is high (e.g., Y X 1600). o would have to e misleading. 4. Using the data calculate XY where X denotes AT and Y denotes GPA. AT GPA AT GPA X Y XY XY Calculate the slope using salary as the dependent variale ( points). XY XX
4 4 6. Calculate the standard error of estimate ( points). Note that Y X Y X 7.6 YY n XY et up and test hypotheses relevant to the question of whether ratings are predictive of salaries; use the F statistic approach and.01 (7 points). H : 0 0 H : 0 A F XX Y X F,1, n F.05,1,
5 5 0.5 F distriution with 1 and 18 df We reject the null hypothesis since calculated F is in the region of rejection. We conclude that ratings are predictive of salaries. 8. The intercept is calculated to e Can a legitimate sustantive interpretation of the intercept e made? Why? No, ecause the range of the ratings is 1 to 5, and the intercept is the estimated conditional mean corresponding to a value of zero for the independent variale.
6 6 9. Calculate and interpret a 90-percent confidence interval for ( points). /, n t Y X XX t /, n t.10/, , 48.7 We are 90% confident that the population slope is etween 4.8 and (The preceding interpretation is satisfactory.) We are 90% confident that the mean difference in salary for secretaries that are onerating point apart is etween 4.8 and (The preceding interpretation is stated in terms of the suject-matter of the prolem and is also satisfactory.) 10. Calculate the residual for a secretary who had a rating of 3.1 and a salary of 750. e Y Yˆ Yˆ ax
7 7 e Y Yˆ The correlation etween performance ratings and iweekly salaries is.78. uppose the reliaility of performance ratings is.81. What is the correlation etween true performance ratings and salary? (Assume salary is measured without error.) ( points.) If salary is measured without error then ryy 1.00 r TXTY rxy rxx ryy In the sample what happens to the average competence rating as more press conferences are viewed? 8.19 ince is positive, in the sample confidence ratings increase as more conferences are viewed. 13. What is the numeric value of the standard error estimate? Y X The conditional variance is equal to the mean square for error. Therefore Y X
8 8 14. Which of the following hypotheses H : 0 0 H : 0 is supported y the results? Use α =.05 and justify your answer with appropriate statistical evidence (3 points). Although the p value is for the t statistic calculated from the slope (i.e. t A ), we can use it ecause the t statistic calculated from the slope equals the t statistic calculated from the correlation: r n 1 r. pro t.0568 We need an upper-tailed p value ecause H A : 0. The sign of r is positive, since its sign is the same as the sign of. ince the sign of r and the hypothesized sign of the population correlation coefficient are the same we compute the upper-tailed p value y ince p 1 pro t , we can reject the null hypothesis; the alternative hypothesis is supported.
9 9 We can also do this prolem y using the calculated t and critical t. Although the calculated t (.01) is for the slope (i.e. t ), we can apply the calculated t it to the correlation hypothesis ecause the t statistic calculated from the slope equals the t statistic calculated from the correlation: r n 1 r. From the printout calculated t equals.01. The critical t is t, t.05, and we reject the null hypothesis. n
10 10 The situation descried in the introduction to 15 and 16 is depicted in the following: Rejected GPA Admitted GRE Figure 1. Hypothetical scatterplot that would have occurred had all applicants een admitted. 15. a The un-standardized slope is not systematically affected y direct selection, so (a) is correct. 16. imple explanation The scatter plot for the selected sample is rounder than is the scatter plot for the entire sample and therefore the correlation is smaller for the selected sample. Because the standardized slope is equal to the correlation coefficient, the standardized slope must also smaller for the selected sample. Complex explanation We know that r 1 n n1 Y X, Y
11 11 Y X is not systematically affected y direct selection, ut Y is reduced y direct selection. Therefore r must e reduced y direct selection and Z r must e smaller in the selected sample. Therefore () is correct. 17. Option () is correct ecause power is lower when the Type I error rate is smaller. In regard to the other options: power increases when (a) the sample size increases and () when a correct directional alternative hypothesis is used. Power is not affected y the choice etween the t test on the regression slope and the t test on the correlation coefficient ecause oth t statistics are equal. 18. d The plot shows the fan-out shape to the residuals so equal conditional variances is violated. In regard to the other options: The most ovious feature in the plot is the fanning out of the residuals, so we should not conclude that the residuals are non-normal. A residual plot cannot tell us whether independence is violated, so () is not correct. There is no relationship etween the residuals and the independent variale, so (c) is not correct. 19. a This follows from the fact that measurement error in either variale attenuates the correlation coefficient. 0. a The width of confidence intervals declines as sample size increases. 1. d Only the correlation coefficient is a scale-free statistic. The size of each of the other statistics changes when the scale of measurement for the variales changes.. c By definition the odds ratio tells us how much the odds are multiplied y when the independent variales is changed one unit (in a descriptive sense).
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