ECON 2331: Economic and Business Statistics 2
Assignment 3 (100 marks; 5%)
To receive full marks, you need to show all your work.
1. Four groups of drivers were selected to test drive three makes (domestic,
Asian, and European) of manual sports cars. The following table shows the average miles per gallon for each group of drivers driving these cars:
(7 marks total)
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|
Drivers
|
|
Automobile
|
1
|
2
|
3
|
4
|
|
Domestic
|
25
|
27
|
20
|
28
|
|
Asian
|
29
|
38
|
24
|
37
|
|
European
|
21
|
28
|
16
|
19
|
a. Use ANOVA and test to see if there is any difference in the miles-per- gallon of the three types of sports car. Let α = .05. (2 marks)
b. Now, due to the differences in their years of experiences, drivers are
treated as four blocks. Test to see if there is any difference in the miles- per-gallon of the three makes of the sports automobiles. Let α = .05. (2 marks)
c. Was the block significant? What does your result imply? (3 marks)
2. The experimental lab of a car company is testing the performance of two
different types of cars with three blends of gasoline. Cars were driven 1000
miles and the gas mileage was recorded in the following table: (10 marks total)
|
|
Blend 1
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Blend 2
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Blend 3
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Asian Cars
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22.6
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19.6
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25.3
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|
20.9
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22.3
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21.6
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|
23.6
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20.5
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20.5
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|
North
American
Cars
|
21.9
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17.0
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24.6
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|
22.0
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17.6
|
22.7
|
|
21.1
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19.2
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20.5
|
a. Clearly state the null and alternative hypotheses. (2 marks)
b. Calculate the test statistic, using α= 0.05. (3 marks)
c. State the conclusion(s). (2 marks)
d. How do you know there exists an interaction term? Interpret the meaning of the interaction term in the context of this experiment. (3 marks)
3. A large annuity company holds many industry group stocks. Among the
industries are banks, business services and construction. Seven companies from each industry group are randomly sampled to test the hypothesis that the mean price per share is the same among industries. The data are:
(10 marks total)
a. State the null and alternative hypotheses. (2 marks)
b. Calculate the MSTR and the MSE. (4 marks)
c. Calculate the test statistic. Using α= 0.05, clearly state the conclusion. (4 marks)
4. The following are the results from a completely randomized design consisting of three treatments: (8 marks total)
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Source of Variation
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Sum of Squares
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Degrees of Freedom
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Mean Square
|
F
|
|
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390.58
|
|
|
|
|
|
158.4
|
|
|
|
|
|
|
23
|
|
|
a. Using α = .05, test to see if there is a significant difference among the
means of the three populations. The sample sizes for the three treatments are equal. (4 marks)
b. If in Part (a) you concluded that at least one mean is different from the others, determine which mean(s) is (are) different. The three sample means are Use Fisherʹs LSD procedure, and let α = .05. (4 marks)
5. In the following table is a partial computer output based on a sample of 21
observations, relating an independent variable (X) and a dependent variable:
(10 marks total)
|
Predictor
|
Coefficient
|
Standard Error
|
|
Constant
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30.139
|
1.181
|
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X
|
-0.2520
|
0.022
|
|
|
|
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SOURCE
|
SS
|
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Regression
|
1,759.481
|
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Error
|
259.186
|
a. Develop the estimated regression line. (2 marks)
b. At α = 0.05, test for the significance of the slope. (2 marks)
c. At α = 0.05, perform. an F-test. (2 marks)
d. Determine the coefficient of determination. (2 marks)
e. Determine the coefficient of correlation. (2 marks)
6. Part of an Excel output relating X (independent variable) andY (dependent variable) is shown in the following tables. Fill in all the blanks marked with letters from A–L. (12 marks)
|
Summary Output
|
|
|
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Regression Statistics
|
|
Multiple R
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A
|
|
R Square
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0.5149
|
|
Adjusted R Square
|
B
|
|
Standard Error
|
7.3413
|
|
Observations
|
11
|
|
ANOVA
|
|
|
|
|
|
df
|
SS
|
MS
|
F
|
Significance F
|
|
Regression
|
C
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D
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E
|
F
|
0.0129
|
|
|
Residual
|
G
|
H
|
I
|
|
|
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Total
|
J
|
1000
|
|
|
|
7. Information regarding a dependent variable y and an independent variable x is provided in the following table: (10 marks total)
|
Σ x = 90
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S (y - y-)( x - x(-)) = -156
|
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Σ y = 340
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Σ(x - x(-))2 = 234
|
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n = 4
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Σ(y - y-)2 = 1974
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SSR = 1704
|
|
a. What are the least squares estimates of b0, b1? (3 marks)
b. What are the sum of squares due to error (SSE), and the total sum of squares (SST)? (4 marks)
c. What is the coefficient of determination? (3 marks)
8. An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of four months was taken. Here are the results of the sample: (17 marks total)
|
Y
|
X
|
|
Monthly Sales
|
Interest Rate (in %)
|
|
22
|
9.2
|
|
20
|
7.6
|
|
10
|
10.4
|
|
45
|
5.3
|
The estimated least squares regression equation is: Ŷ= 75.061 - 6.254X
a. Obtain a measure of how well the estimated regression line fits the data. (3 marks)
b. You want to test to see if there is a significant relationship between the interest rate and monthly sales at the 1% level of significance. State the null and alternative hypotheses, and test the hypotheses. (4 marks)
c. Construct a 99% confidence interval for the average monthly sales for all months with a 10% interest rate. (5 marks)
d. Construct a 99% prediction interval for the monthly sales of one month with a 10% interest rate. (5 marks)
9. Quality of a special type of oil is measured in API gravity degrees: the higher the degrees API, the higher the quality. The following table is produced by an expert in the field who believes that there is a relationship between quality and price per barrel: (16 marks total)
a. Estimate the regression equation and determine the predicted values of y. (3 marks)
b. Use the predicted values and the actual values of y to calculate the residuals. (3 marks)
c. Plot the residuals (vertical axis) against the predicted values (horizontal axis). (2 marks)
d. Does it appear that heteroscedasticity is a problem? Explain. (2 marks)
e. Draw a histogram of the residuals. Does it appear that the errors are normally distributed? Explain. (3 marks)
f. Use the residuals to identify possible outlier(s). (3 marks)