COR1-GB.1305.03
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 annual data from the International Monetary Fund on the
G7 countries, spanning the years 2002-2011 (n=70).
The response variable (y) is the inflation rate (percent)
The two explanatory variables are:
X1=Percent Change in GDP (from previous year)
X2=Unemployment Rate (percent of total labor force)
Figure 1 provides a fitted line plot for the regression of y on X1.
Figure 2 provides a fitted line plot for the regression of y on X2.
Fig 1: Fitted Line Plot for Inflation vs. ChangeGDP
Inflation = 1.575 + 0.1514 ChangeGDP
S
R-Sq
R-Sq(adj)
4
1.12248
7.9%
6.5%
Inflation
3
2
1
0
-1
-2
-7.5
-5.0
-2.5
0.0
ChangeGDP
2.5
5.0
1) Consider a simple regression of y on X1. The corresponding Minitab output is given
below.
Regression Analysis: Inflation versus ChangeGDP
The regression equation is
Inflation = 1.58 + 0.151 ChangeGDP
Predictor
Constant
ChangeGDP
Coef
1.5754
0.15136
S = 1.12248
SE Coef
0.1539
0.06282
R-Sq = 7.9%
T
10.24
2.41
P
0.000
0.019
R-Sq(adj) = 6.5%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
68
69
SS
7.315
85.677
92.993
MS
7.315
1.260
F
5.81
P
0.019
A) Is there a strong linear relationship between Inflation and ChangeGDP? Explain.
(2 points).
B) Is there evidence at the 1% level of significance of a positive linear relationship
between Inflation and ChangeGDP? (4 points).
C) Provide an interpretation of the estimated slope in the model. (2 points).
D) Provide an interpretation of the value of S in the Minitab output, in terms of
inflation and GDP. (2 points).
Fig 2: Fitted Line Plot for Inflation vs. Unemployment
Inflation = 0.9394 + 0.1157 Unemployment
S
R-Sq
R-Sq(adj)
4
1.15078
3.2%
1.7%
Inflation
3
2
1
0
-1
-2
3
4
5
6
7
8
Unemployment
9
10
11
2) Consider a simple regression of y on X2. The corresponding Minitab output is given
below.
Regression Analysis: Inflation versus Unemployment
The regression equation is
Inflation = 0.939 + 0.116 Unemployment
Predictor
Constant
Unemployment
Coef
0.9394
0.11568
SE Coef
0.5658
0.07764
S = 1.15078
R-Sq = 3.2%
T
1.66
1.49
P
0.101
0.141
R-Sq(adj) = 1.7%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
68
69
SS
2.940
90.052
92.993
MS
2.940
1.324
F
2.22
P
0.141
A) In 2011, Germany had an unemployment rate of 6.558 percent. What is the fitted
value for this data point? (2 points).
B) Can we think of the fitted value in A) as a forecast for inflation in Germany in
2012? Explain. (4 points).
C) Based on the Minitab output here and in question 1), if you could use just one of
Unemployment or Change in GDP as an explanatory variable, which one would
you use and why? (4 points).
3) Here are the results for a multiple regression of y on X1 and X2.
Regression Analysis: Inflation versus ChangeGDP, Unemployment
The regression equation is
Inflation = 0.526 + 0.169 ChangeGDP + 0.145 Unemployment
Predictor
Constant
ChangeGDP
Unemployment
Coef
0.5264
0.16897
0.14541
SE Coef
0.5620
0.06225
0.07504
S = 1.10041
R-Sq = 12.8%
T
0.94
2.71
1.94
P
0.352
0.008
0.057
R-Sq(adj) = 10.2%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
2
67
69
SS
11.862
81.130
92.993
MS
5.931
1.211
F
4.90
P
0.010
A) Based on the output here and in Question 2), when does Unemployment seem to be
more helpful for describing Inflation: when Unemployment is used by itself, or
when Unemployment is used in conjunction with ChangeGDP? (7 points).
B) Show how S=1.10041 above is obtained from other numbers in the Minitab output.
(3 points).
4)
A) Provide an interpretation of the coefficient of Unemployment in the multiple
regression. (5 points).
B) Construct a 95% confidence interval for the true coefficient of Unemployment in
the multiple regression. (5 points).
5) The n=70 values of ChangeGDP used in the regressions above had a sample mean of
1.2008 and a sample standard deviation of 2.1512.
A) Is there evidence at the 5% level of significance that the population mean of
ChangeGDP is positive? (5 points).
B) For a t-test to be valid, the observations need to be independent. Is it reasonable to
assume that the observations of ChangeGDP are independent? (5 points).
6) Based on the multiple regression above, Figure 3 plots the residuals vs. the year of
observation. Does Figure 3 reveal any problems with the model?
Fig 3: Resids vs Year For Multiple Regression
3
2
RESI1
1
0
-1
-2
-3
2002
2003
2004
2005
2006
2007
Year
2008
2009
2010
2011
7)
A) In the sampling lab, 54 class members each selected a sample of size 5 from a
population with mean μ=11.24 and standard deviation 9.11. You each calculated a
sample mean for the five values in your sample. The average of all of the x
values was 13.09, which is larger than the population mean by 1.85. Assuming
that the samples were taken at random (and that the x values are independent of
each other) what is the probability that the average of the x values would differ
by at least 1.85 in either direction from the true population mean? (Hint: You may
need to get an approximation to this probability: If you are off the chart in Table
5, try using Table 6 to get a rough answer.) (7 points).
B) Provide an interpretation of the probability you computed in Part A). Focus on the
question as to whether the samples were indeed independent random samples.
(3 points).
8) Suppose you run a simple regression in Minitab, based on a data set with n=10. If
Minitab’s p-value corresponding to the estimated slope in the regression is .05, then
what is the value of the t-statistic for the slope? (Assume that the estimated slope is
negative.)
9) If P(AB) > P(A)+P(B) then show that A, B cannot be complements of each other.
10) A sample of size 15 from a normal population yields a sample mean of 3.1 and a
sample standard deviation of 5.2. Test the null hypothesis H 0 :   2 versus the
alternative hypothesis H A :   2 at the 5% level of significance.