(a) The median test score on a Stat 139 exam was 87 (out of max of 100). Provide a reasonable value for the mean for these test scores. Explain. (b) In the multiple regression model based on OLS with no missing data (no explanations needed): i. True or False: R" cannot be negative ii. True or False: R" is the percent of variability in the response variable associated with the predictors ili. True or False: R equals the square of the correlation coefficient r iv. True or False: R may decrease when an additional explanatory variable is added (c) You were given two samples of the same size. The samples were paired. However, you mistakenly treated them as independent data and ran a 2-sample f-test. Which of the following was (were) affected by this mistake. Briefly explain your answers. i. The difference in sample means ii. The standard error of the difference in sample means iii. The significance level of the test iv. The reference distribution of the f-test v. Would you have understated or overstated the p-value? (d) After transforming two groups of data. X and Y', via the square root function, the histograms both looked symmetrically distributed. A confidence interval for the true mean difference on the square root scale of (ox - my) was calculated to be (3, 11). Which of the following can be concluded about the distributions of X and Y on the original scale? Briefly explain your alewers. i. The result is not statistically significant. ii. The means of X and Y' are significantly different. ili. The medians of X and Y are significantly different. iv. The confidence interval for the ratio of medians (moms ) is estimated to be (4. 100). (e) Let X1. ...Xox ~ N(ox."'). Y..... Y, ~ N(oy. 57), and 21..... Zng ~ N(pz.07) with all observations indpendent. i. Determine the sampling distribution of 83 = Li-L next ny + nz- 3 ii. You'd like to test Ho : Ax = py. Determine the sampling distribution of (if the null hypothesis is true): X - Y T=