Problem session #3
Q.1
The t-statistic is calculated by dividing
a. the OLS estimator by its standard error.
b. the slope by the standard deviation of the explanatory variable.
c. the estimator minus its hypothesized value by the standard error of the estimator.
d. the slope by 1.96.
Q.2
If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal
distribution, you can
a. reject the null hypothesis.
b. safely assume that your regression results are significant.
c. reject the assumption that the error terms are homoskedastic.
d. conclude that most of the actual values are very close to the regression line.
Q.3
Consider the following regression line: TestScore  698.9  2.28  STR . You are told that the t-statistic
on the slope coefficient is 4.38. What is the standard error of the slope coefficient?
a. 0.52
b. 1.96
c. -1.96
d. 4.38
Q.4
One of the following steps is not required as a step to test for the null hypothesis:
a.
b.
c.
d.
Q.5
compute the standard error of  1 .
test for the errors to be normally distributed.
compute the t-statistic.
compute the p-value
Finding a small value of the p-value (e.g. less than 5%)
a. indicates evidence in favor of the null hypothesis.
b. implies that the t-statistic is less than 1.96.
c. indicates evidence in against the null hypothesis.
d. will only happen roughly one in twenty samples
Q.6
True or False. Explain
a. Best of BLUE describes maximum variance.
b. hat 2 = .50 our null hypothesis is 2 = 0 You must reject the null based on these two facts. C
c. Our obtained (calculated) t equals 19, the beta in question is statistically not significant?
d. The method of least squares maximizes the explained variation.
Analytical Questions
Q.1
SCENARIO: The marketing sub-committee for a consortium of dealers of American-made luxury automobiles has
hired your consulting firm to tell them about the determinants of demand for their products. For a random
sample of 17 dealerships in different neighborhoods, you collect data on the average number of cars sold per
month (carsi), the median age of the population in the same zipcode as the dealership (agei), the median
household income (in thousands of dollars) from all sources in the same zipcode (inci), median price of luxury
cars stocked at the dealership (in thousands of dollars) (pricei), and distance from the nearest foreign luxury car
dealership (disti). The statistical analyses you perform are given in the Exhibit below
a.
Using the information under descriptive statistics and the regression output, fill in the 7 blanks marked
with XXXX under Regression 1 in the Exhibit below.
Based on Regression 1, what is the verbal interpretation of the estimated regression equation? Is the
intercept meaningful? Why or why not?
Based upon the simple regression results the Regression 1, do cars appear to be inferior (as opposed to
normal) goods? Explain how you have reached this conclusion.?
Based upon the simple regression results in Regression 1, does income appear to have any significant
relationship with cars? Test the hypothesis at the 5% level.
Based on Regression 1, what level of monthly average car sold is expected for a median household
income of $50,000 in the same zipcode? Give the formula and BUT DO NOT CALCULATE how a 95%
confidence interval for this prediction would be constructed
The chairperson for the consortium says "I took Econ 1 and I know that demand curves slope
downwards from left to right. I don't think much of your skills as an economist if the demand curve you
estimate in Regression 2 is not characterized by a negative slope." Based upon Regression 2 in the
Exhibits, is it possible that there is a downward sloping demand curve for these cars? Explain how you
have reached this conclusion
b.
c.
d.
e.
f.
Q.2
Consider the regression model Y = A+BX+, where Y is the aggregate investment for all firms over a number of
past periods (in billions of dollars) and X is the composite price index of common stocks over the given period.
You wish to estimate the regression based on 50 sample data. A spreadsheet calculation with these 50
observations gave the following results:
i = 13778.4
Yi = 1778.3
iYi = 412329.68
i2 = 4425379.14
Yi2 = 74711,37
a.
Determine the least square regression equation. (5)
b.
Calculate the proportion of investment level that is explained by the price index? (3)
c.
Test the hypothesis that there is no significant relationship between aggregate investment and the
composite price index at the 5% level
Q.3
Given the following estimated value for the following regression (standard errors of estimated coefficients are in
brackets)
Pwidth = 2.5 - .77height
( .77) (.33)
Test the hypothesis that I think there is a direct relationship between height and Pwidth: hint: test if B1 =1
EXHIBITS
|_* Descriptive Statistics
NAME
N MEAN
AGE
17 55.041
INC
17 186.94
CARS
17 6.2659
PRICE
17 53.941
DIST
17 11.414
ST. DEV VARIANCE MINIMUM MAXIMUM
13.538
183.29
29.714
70.265
38.230
1461.6
100.00
250.00
3.1415
9.8688
1.7955
12.357
2.5365
6.4338
49.000
59.000
6.5447
42.833
2.0830
21.814
Regression 1
|_ols cars inc
R-SQUARE = XXXXX
STANDARD ERROR OF THE ESTIMATE-SIGMA = XXXXX
SUM OF SQUARED ERRORS-SSE= 86.037
MEAN OF DEPENDENT VARIABLE = XXXXX
∑ 𝑌 2 = 𝑋𝑋𝑋𝑋𝑋
ANALYSIS OF VARIANCE SS
DF
MS
REGRESSION
71.865
1.
71.865
ERROR
86.037
15.
5.7358
TOTAL
157.90
XXXXX.
9.8688
VARIABLE ESTIMATED STANDARD T-RATIO
NAME COEFFICIENT ERROR 15 DF P-VALUE
INC
0.055436 0.01566 XXXXX 0.003
CONSTANT -4.0973 XXXXX -1.373 0.190
|_* Regression 2
|_ols cars price
R-SQUARE = 0.1185 R-SQUARE ADJUSTED = 0.0597
VARIANCE OF THE ESTIMATE-SIGMA**2 = 9.2795
STANDARD ERROR OF THE ESTIMATE-SIGMA = 3.0462
SUM OF SQUARED ERRORS-SSE= 139.19
MEAN OF DEPENDENT VARIABLE = 6.2659
ANALYSIS OF VARIANCE - FROM MEAN
SS
DF
MS
REGRESSION
18.709
1.
18.709
ERROR
139.19
15.
9.2795
TOTAL
157.90
16.
9.8688
VARIABLE ESTIMATED STANDARD T-RATIO
NAME COEFFICIENT ERROR 15 DF P-VALUE
PRICE 0.42632 0.3002
1.420 0.176
CONSTANT -16.730 16.21 -1.032 0.318