QUESTIONS 1-8 DEAL WITH THE FOLLOWING SITUATION: Stock Prices, Y, are assumed to be affected by the annual dividend rate, X1, and annual return on equity, X2. A first order regression was fit to the data and the following analysis resulted.
Sum of Mean
Source DF Squares Square F Value Prob
Model 2 1148.64291 574.32145 57.887 0.0001
Error 25 248.03566 9.92143
Total (Adjusted) 27 1396.67857
Square Root MSE 3.14983 R-square 0.8224
Dep Mean 22.10714
Parameter Estimates
Coefficient Standard T for H0:
Variable Estimate Error B=0 Prob
INTERCEPT -15.830443 4.67316808 -3.388 0.0023
X1 12.276010 1.19702953 10.255 0.0001
X2 0.609433 0.28306125 2.153 0.0412
Actual Predicted 95% LCL 95%UCL 95% LCL 95% UCL
Obs X1 X2 Var Y Value Mean Mean Individual Individual
29 2.96 17.1 . 30.9279 28.4359 33.4198 23.9785 37.8772
1. I am 95% confident that the average stock price for all stocks with a dividend rate of 2.96 and an equity return of 17.1 falls in the range
A. 35.00 to 30.92
B. 20.95 to 23.74
C. 28.44 to 33.42
D. 15.71 to 28.98
E. 23.98 to 37.88
2. What is the p-value for testing for the effect of equity?
A. 2.153
B. 0.0001
C. 0.0412
D. 0.05
E. 0.8224
3. What percent of the sample variability in stock prices can be attributed to variation in the dividend rate and the return on equity.
A. 78.14
B. 82.24
C. 78.95
D. 80.82
E. 89.27
4. The rejection region for testing that equity and/or dividend rate is useful for estimating the mean stock price is
A. F > F(2, 25, 0.05)
B. |t| > t(2, 25, 0.025)
C. t > t(25, 0.025) or t < -t(25, 0.025)
D. F < F(25, 0.05)
E. F > F(25, 0.05)
5. If there is interaction between annual rate of dividend and return on equity then
A. the stock price interacts with annual rate of dividend
B. the change in the mean stock price associated with each additional point in the dividend rate is a linear function of the annual return on equity.
C. the slope of annual rate of dividend is a function of the stock price
D. the mean stock price depends on annual rate of dividend and/or the number of equity (only true for interaction model)
E. annual rate of dividend is correlated with the return on equity.
6. What is your conclusion after testing b2 = 0 versus b2 ≠ 0?
A. At alpha=0.05, we can say that after adjusting for the dividend rate , the return on equity does help predict the stock price.
B. At alpha=0.05, we can not say that the dividend rate does help predict the stock price.
C. At alpha=0.05, we can say that the dividend rate does help predict the stock price.
D. At alpha=0.05, we can say that after adjusting for the dividend rate, the return on equity does not help predict the stock price.
E. At alpha=0.05, we can not say that after adjusting for the dividend rate, the return on equity does help predict the stock price.
7. What is the estimate of the typical sample error when trying to predict stock price with the dividend rate and return on equity?
A. 3.14983
B. 2.31635
C. 0.82240
D. 4.67317
E. 1.19703
Answers:
1. C Observation 11, third and fourth columns from left
2. C p-value:
3. B definition of r-square
4. A F(k, n-k-1)
5 B interaction interpretation
6. A testing equity - p-value = 0.0412 < 0.05 reject H0
7. A root MSE is the standard error of y given x1 and x2
QUESTIONS 8-14 DEAL WITH THE FOLLOWING SITUATION: A collector of antique grandfather clocks believes that the price received for the clocks, Y, at an antique auction increases with the age of the clocks, X1, and with the number of bidders, X2. A first order regression was fit to a random sample of 32 clocks with the following analysis resulted.
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pro
Model 2 4277159.7034 2138579.8517 120.651 0.0
Error 29 514034.51534 17725.32812
C Total 31 4791194.2188
Root MSE 133.13650 R-square 0.8927
Dep Mean 1327.15625 Adj R-sq 0.8853
Parameter Estimates
Coefficient Standard T for H0:
Variable Estimate Error Parameter=0 Prob
INTERCEP -1336.722052 173.35612607 -7.711 0.0001
X1 12.736199 0.90238049 14.114 0.0001
X2 85.815133 8.70575681 9.857 0.0001
8. The estimated slope of number of bidders is:
A 85.815133
B 0.90238049
C 0.0001.
D 12.736199
E 14.114
9. The true slope parameter of age is interpreted as:
A The change in the mean auction price for each additional year of age.
B The change in the auction price for each additional year of age.
C The change in the mean auction price for each additional year of age when the number of bidders is held constant,
D The mean auction price given the age, holding the number of bidders constant.
E The change in the estimated mean auction price for each additional year of age.
10. The test statistic value for testing the utility of the model is:
A 3.33
B 120.651
C 14.114.
D 133.1365
E 0.8927
11. The rejection region for testing that increases in the mean auction price of the clock will be associated with increases in the number of bidders ( age held constant) is
A t > 2.042 or t < -2.042
B t > 2.045
C t > 1.699 or t < -1.699
D t > 1.699
E t > 1.697
12. The probability of saying that "the mean auction price is different for clocks one year apart in age (holding number of bidders constant)" when actually there is no difference is called
A a Type II error
B the p-value
C the slope.
D the power of the test
E alpha
13. The degrees of freedom of the estimated variation in the auction prices for all clocks of the same age and number of bidders is:
A 1
B 31
C 30
D 29
E 2
14. If there is interaction between age and number of bidders then
A the auction price interacts with age
B the change in the mean auction price associated with each additional bidder is a linear function of the age.
C the slope of age is a function of the auction price.
D the mean auction price depends on age and/or the number of bidders (only true for interaction model)
E age is correlated with the number of bidders.
ANSWERS
8. A
9. C
10. B
11. D
12. E
13. D
14. B
QUESTIONS 15-23 DEAL WITH THE FOLLOWING SITUATION: The expected sales of a product, Y, in a city are assumed to be affected by the per capita discretionary income (PCDI), X1, and the population of the city, X2. A first order model was fit to a random sample of 15 cities
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 2 53844.71643 26922.35822 5679.466 0.0001
Error 12 56.88357 4.74030
Total 14 53901.60000
Root MSE 2.17722 R-square 0.9989
Dep Mean 150.60000 Adj R-sq 0.9988
Parameter Estimates
Parameter Standard T for H0:
Variable Estimate Error Parameter=0 Prob
INTERCEP 3.452 2.430 1.420 0.1809
POP 0.496 0.006 81.924 0.0001
INCOME 0.009 0.001 9.502 0.0001
Actual Pred Lower95% Upper95% Lower95% Upper95%
Obs POP INCOME Mean Mean Predict Predict
16 220 2500 . 135.6 134.1 137.1 130.6 140.5
17 375 3500 . 221.7 219.8 223.6 216.5 226.8
15. The null hypothesis for the test of model usefullness is interpreted as
A. PCDI and Population size are not linearly related.
B. Mean sales is not a linear function of PCDI and Population size.
C. PCDI and Population size do not vary.
D. PCDI and Population size do not help predict the sales.
E. PCDI and Population size do help predict the sales.
16. What is the test statistic value when testing that both coefficients
are equal to zero?
A. 2.17722
B. 81.924 + 9.502 = 90.426
C. 0.1809
D. 0.9988
E. 5679.466
17. What is the interpretation of the prediction interval for observation
17?
A. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $219,800 and $223,600.
B. With 95% confidence we can say that all cities with 375,000 people and
a PCDI of $3,500 would have mean sales between $219,800 and $223,600.
C. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $67,300 and $292,300.
D. With 95% confidence we can say that all cities with 375,000 people and
a PCDI of $3,500 would have mean sales between $216,500 and $226,800.
E. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $216,500 and $226,800.
18. When testing the alternative hypothesis that b1 is less than zero,
what is your conclusion?
A. Since p-value =0.0001, we can say that when holding PCDI constant
increases in population size is associated with decreases in mean sales.
B. Since p-value > 0.05, we can say that increases in population size is
associated with decreases in mean sales.
C. Since the test statistic value is 9.502, we can not say that when
holding PCDI constant increases in population size is associated with
increases in mean sales.
D. Since p-value < 0.05, we can say that when holding city size constant,
PCDI does help estimate mean sales.
E. Since p-value > 0.05, we can not say that when holding PCDI constant
increases in population size is associated with decreases in mean sales.
19. What would be the rejection region value when testing that the change
in the mean sales with each one dollar increase in the PCDI depends on the
number of people in the city. Reject Ho if
A. |t| > t(12, 0.025)
B. F > F(2, 12, 0.05)
C. chi-squared > chi-squared( 12, 0.05)
D. t > t(11, 0.025) or t < -t(11, 0.025)
E. F > F(3, 11, 0.05)
20. What is the meaning of the confidence interval for the mean value of Y
given X1=375 and X2=3500?
A. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $67,300 and $292,300.
B. With 95% confidence we can say that all cities with 375,000 people and
a PCDI of $3,500 would have mean sales between $216,500 and $226,800.
C. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $216,500 and $226,800.
D. With 95% confidence we can say that a city with 375,000 people and a
PCDI of $3,500 would have sales between $219,800 and $223,600.
E. With 95% confidence we can say that all cities with 375,000 people and
a PCDI of $3,500 would have mean sales between $219,800 and $223,600.
22. For all cities with the same PCDI, what is the estimated change in the
expected sales when the population of the city increases by one?
A. 99.89% increase
B. 99.88% increase
C. $ 2.177
D. $ 9
E. $ 496
22. Does interaction of X1 and X2 imply that there is correlation between
X1 and X2?
a. Yes.
b. No.
23. What is the value for the multiple coefficient of determination?
A. 2.177
B. 0.0001
C. 9.502
D. 1.4457
E. 99.89%
Answers
------
15. d
16. e
17. e
18. e
19. d
20. e
21. e
22. b
23. e
Question 24-27 deal with the following situation: An instructor of BUSA 5325
wants to know if the exam scores of the third exam can be predicted from
the exam scores of the first two exams. The numeric scores of the second
and third exam are available. However, the first exam seems not to have a
linear relationship with exam-3 and has been changed to a categorical
variable. The variables are
Y = Exam score on the third exam
X1a = 1 if student made an A on exam 1
0 if not
X1b = 1 if student made a B on exam 1
0 if not
X2 = Exam score on the second exam
The following model is proposed:
E(Y) = b0 + b1X1a + b2X1b + b3 X2
A random sample of 32 of the students was selected and the analysis of
variance report is below.
PARALLEL (NO-INTERACTION)MODEL - Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 2347.91005 782.63668 7.308 0.0009
Error 28 2998.55870 107.09138
C Total 31 5346.46875
Root MSE 10.34850 R-square 0.4392
Dep Mean 77.28125
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 47.043655 12.71357992 3.700 0.0009
X1a 1 9.801066 4.17220766 2.349 0.0261
X1b 1 -5.364949 4.02638082 -1.332 0.1935
X2 1 0.356597 0.15862042 2.248 0.0326
Standardized Variance
Variable DF Estimate Tolerance Inflation
INTERCEP 1 0.00000000 . 0.00000000
X1a 1 0.37838587 0.77202700 1.29529148
X1b 1 -0.20590050 0.83883060 1.19213582
X2 1 0.36907595 0.74317755 1.34557349
24. Based on the above F value 7.308 and its p-value, the following conclusion can be made: At alpha = 0.05
A. We can say that the average exam-3 scores is affected by either the
exam-2 scores or the exam-1 categories.
B. We can say that the average exam-3 scores differ among the exam 1