B01.1305
FINAL EXAM
This is the question sheet. There are 10 questions, each worth 10 points. Please write all answers in the answer book, and justify your answers. Good Luck!
In questions 1-6, we consider data on executive pay for the top 50 CEOs for 2007.
Here, "top 50" refers to CEO compensation.
The y-variable is the 2007 company performance rank. (We won't go into details on how
company performance was measured, but it's a ranking with respect to a largerset of companies). Rank 1 was the highest performing company in our dataset, Conoco Phillips. Rank 188 was the lowest performing company in our dataset, Winn-Dixie Stores.
The first explanatory variable is the Compensation Rank for the CEO. Rank 1 went to John A. Thain (MerrillLynch), whose total compensation was $83.8 Million. Rank 50 went to Richard T. Clark (Merck),with a total compensation of $2.9 Million.
The second explanatory variable is the base salary for the CEO.
1)
A)Write down the simple linear regression model for company performance rank in terms of compensation rank. For each quantity appearing in the model, state whether it is known (observed) or not. Please note that I am asking here about the true regression model, not the particular fitted model that we got for the given data set. (5 points).
B)What values of the true slope coefficient in the model would tend to suggest that CEOs earnedtheir compensation? Explain. (5 points).
Figure 1 plots the company performance rank versus the compensation rank. The corresponding Minitab regression output is given below.
Regression Analysis: Company Performance Rank versus Compensation Rank
The regression equation is
Company Performance Rank = 34.5 + 0.900 Compensation Rank
Predictor Coef SE Coef T P
Constant 34.51 12.94 2.67 0.010
Compensation Rank 0.9000 0.4416 2.04 0.047
S = 45.0652 R-Sq = 8.0% R-Sq(adj) = 6.0%
Analysis of Variance
Source DF SS MS F P
Regression 1 8435 8435 4.15 0.047
Residual Error 48 97482 2031
Total 49 105916
2) Based on the information given above, answer these questions.
A) Is there evidence at the 5% level of significance that the CEOs earned their compensation? (2 Points).
B)Is there evidence at the 4% level of significance that the CEOs earned their compensation?(2 Points).
C) Do Figure 1 and the regression output above suggest that (CEO) compensation rank is a good predictorof company performance rank?(2 Points).
D)Predict the company performance rank given a CEO compensation rank of 10. Give a point prediction.(2 Points).
E)Alan G. Lafley of Proctor & Gamble had compensation rank 10. The residual for Proctor & Gamble was −29.51. What is the performance rank for Proctor & Gamble? (2 Points).
3)Is there evidence to contradict the hypothesis that a one-point improvement in the CEO compensationrank leads to a one-point improvement in the expected company performance rank? Use the 5% levelof significance.
We now consider a multiple regression of company performance rank on (CEO) compensation rank and (CEO) base salary in US Dollars. The Minitab regression output is given below, and the plot of residualsvs. fits is given in Figure 2. Note that some of the values in the analysis of variance tablebelow were intentionally left blank.
Regression Analysis: Company Performance Rank versus Compensation, Base Salary
The regression equation is
Company Performance Rank = 44.3 + 0.864 Compensation Rank
- 0.000006 Base Salary
Predictor Coef SE Coef T P
Constant 44.29 16.14 2.74 0.009
Compensation Rank 0.8644 0.4429 1.95 0.057
Base Salary -0.00000619 0.00000611 -1.01 0.316
S = 45.0526 R-Sq = 9.9% R-Sq(adj) = 6.1%
Analysis of Variance
Source DF SS MS F P
Regression * * * * 0.086
Residual Error * * *
Total 49 105916
4)
A) Interpret the coefficient for base salary in the Minitab output above. (3 points).
B)What is the appropriate alternative hypothesis for the true coefficient of base salary?(3 points).
C) With respect to the alternative hypothesis in B), is the coefficient of base salary
statistically significant at the 5% level of significance? (2 points).
D) Does the residuals vs. fits plot in Figure 2 suggest any problems with the multiple regressionmodel? (2 points).
5)
A) Interpret the p-value for the F-statistic above. (3 points).
B) Does the p-value for the F-statistic provide information on whether compensation rankis a useful variable for predicting company performance rank? If not, why not? If so, what information does it provide? (4 points)
C) Does the p-value for the F-statistic contradict the p-value for the coefficient ofcompensation rank in the simple regression we performed earlier? (3 points)
6) Some of the entries in the analysis of variance table above were intentionally left blank.Based on the multiple regression output providedabove (the full output, not limited to the analysis of variance part), compute the value of the F-statistic.
7)Suppose that you have a null hypothesis , an alternative hypothesis , and suppose that for a given data set you obtaina corresponding p-value of .023. For this situation, consider the following statement: "The null hypothesis would be rejected 2.3% of the time". If this statement is true, explain why. If the statement is false, explain why.
8)In hypothesis testing, if the null hypothesis can be rejected at the 1% level of significance,does this imply that it can also be rejected at the 5% level of significance? Explain.
9) The household net savings rate for the United States (annual data from 1996 to 2005) averaged to1.72%, with a sample standard deviation of 2.334%.
A) If this data set is considered to be a random sample drawn independently
from a population, what would the population mean represent? (4 points).
B) Based on the assumption above, construct a 95% confidence interval for the population mean, and provide an interpretation. (4 points)
C) Explain why we might be suspicious of the assumption that the values obtained in the sample here can be considered to be independent of each other. (2 points). The actual data values aregiven below.
Year / Household Net Savings Rate1996 / 5.9
1997 / 5.8
1998 / 2.1
1999 / 0.5
2000 / 0.5
2001 / 2.0
2002 / 0.5
2003 / 0.7
2004 / -0.7
2005 / -0.1
10) If X and Y are independent standard normal random variables, what is the probability
that XY is positive? Justify your answer.