Name:_______________________________
Student #:____________________________
Section #: ____________________________
BUEC 333 MIDTERM
There is a formula sheet attached at the end of the exam and some blank pages for your rough work.
Use the bubble sheet to record your answers for the multiple choice questions and write your answers
to the short answer questions in the space provided.
Multiple Choice (1 points each)
1.) Suppose that on any given trading day, a stock’s price can increase, decrease, or remain
unchanged. Likewise, on any given day, the weather can be sunny or cloudy. You have the
following information about the joint distribution of weather and stock price changes. It is
sunny today. What is the probability of a price increase?
Sunny
Cloudy
Decrease
3/20
1/10
Unchanged
3/10
1/5
Increase
1/10
a) 1/4
b) 3/10
c) 3/16
d) 3/20
e) none of the above
2.) If the covariance between two random variables X and Y is zero then
a) X and Y are independent
b) Knowing the value of X provides no information about the value of Y
c) E(X) = E(Y) = 0
d) a and b are true
e) none of the above
3.) The value of X1 is an unbiased estimator of the sample mean. Now consider, the estimator
(X1 + X2). Is this another unbiased estimator of the population mean?
a) Yes
b) No
c) Depends on the value of X1
d) Depends on the value of (X1 + X2)
e) none of the above
4.) Suppose you have a random sample of 20 observations from a normal distribution with
mean = 10 and variance = 2. The sample mean (x-bar) is 8 and the sample variance is 3. The
sampling distribution of x-bar has
a) mean 8 and variance 3
b) mean 8 and variance 0.3
c) mean 10 and variance 0.2
d) mean 10 and variance 0.1
e) none of the above
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5.) The law of large numbers says that:
a) the sample mean is a biased estimator of the population mean in small samples
b) the sampling distribution of the sample mean approaches a normal distribution as
the sample size approaches infinity
c) the behaviour of large populations is well approximated by the average
d) the sample mean is an unbiased estimator of the population mean in large samples
e) none of the above
6.) The power of a test is the probability that you:
a) reject the null when it is true
b) fail to reject the null when it is false
c) reject the null when it is false
d) fail to reject the null when it is true
e) none of the above
7.) Which of the following is not a linear regression model:
a) Yi    X i  X i2   i
b) Yi   0  1 X i
2
 i
c) log( Yi )   0  1 log( X i )   i
d) Yi   0  1 log( X i )   i
e) all of the above
8.) In the linear regression model, the least squares estimator
a) minimizes the sum of squared residuals
b) is unbiased
c) is most efficient among the class of linear estimators
d) maximizes the value of R2
e) all of the above
9.) In the linear regression model, the degrees of freedom
a) is equal to the number of observations (n) minus k minus 1
b) affects the precision of the coefficient estimates
c) does not affect the value of the coefficient estimates
d) all of the above
e) none of the above
10.) In the Capital Asset Pricing Model (CAPM),
a) R2 is meaningless
b) β is greater than one
c) α is less than zero
d) β measures the sensitivity of the expected return of a portfolio to specific risk
e) β measures the sensitivity of the expected return of a portfolio to systematic risk
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Short Answer (10 points each – show your work!)
1.) Consider the case of a uniformly distributed random variable where each outcome
(1, 2, 3, 4) has an equal chance of occurring.
a) What is the population mean and variance of this random variable?
b) Suppose that a random number generator provides the following sequence of numbers,
2-1-4-1. What is the mean and variance of this sample?
c) What is the sampling distribution of the sample mean calculated above? Provide a verbal
interpretation of the sampling distribution of the sample mean.
d) Compute the value of the t-statistic for testing the null hypothesis that μ = 2.5.
e) The critical value for a t distribution with 3 degrees of freedom and a 0.20 level of
significance in the presence of a two-sided alternative is equal to 1.638. Can you
reject the null hypothesis that μ = 2.5 at the 20% level of significance? What about at
the 10% level of significance?
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2.) Suppose that on any given day, the length of time spent waiting to board the gondola at
Whistler Mountain can be short, medium, or long. Likewise, on any given day, there can be
fresh snow on the mountain or not. You have the following information about the joint
distribution of snowfall and waiting times:
Fresh Snow
No Fresh Snow
Short Wait
1/20
1/5
Medium Wait
1/10
3/10
Long Wait
5/20
a) What is the joint probability that there is no fresh snow and the wait time is long?
b) If a short wait is one minute, a medium wait is ten minutes, and a long wait is twenty
minutes, what is the expected waiting time on any given day?
c) There is NO fresh snow today. What is the probability of a long wait?
d) Explain the difference between your answer to part a) and your answer to part c).
e) Are wait times and weather independent? Explain.
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3.) Suppose we have a linear regression model with two independent variables and a
constant:
Yi = β0 + β1X1i + β2X2i + εi
Verbally, explain the steps necessary to derive the least squares estimator, providing as much
intuition as possible for each step.
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4.) Suppose that in a random sample of 100 SFU graduates, you estimate the following
regression equation:
Yi   0  1 X 1i   2 X 2i   3 X 3i   i
Yi = Hourly wage (in dollars) at first full-time job
X1i = Cumulative GPA at graduation
X2i = Number of years to complete SFU degree
X3i = Usual number of hours worked off-campus per week while a student
You estimate the model in EViews and obtain the following:
ˆ0  10 ˆ1  5 ˆ2  1 ˆ3  0.2 R 2  0.3
a) Interpret the estimates ˆ and ˆ .
0
1
b) Compute the predicted wage of a graduate with a GPA of 3.0, who completes their degree
in 5 years, and who worked 10 hours per week off-campus.
c) What is the value of this student’s residual if their hourly wage is $25? What does the
value of this student’s residual tell you (that is, interpret its value)?
d.) If a student from the same graduating class had a GPA of 4.0 and took only four years to
complete their degree, how many hours per week do you predict they would need to
have worked as a student in order to make that same $25 hourly wage?
e.) Are there any important variables missing from the regression above? If so, name one
which it should be easy for you and/or the university to collect data on.
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Useful Formulas:
E( X ) 

k

  x  
Var ( X )  E  X   X  
p i xi
i 1
2
k
i
X
2 pi
i 1
Pr( X  x, Y  y)
Pr(Y  y | X  x) 
Pr( X  x)
k
Pr( X  x)   Pr X  x, Y  yi 
i 1
k
m
E Y | X  x    yi PrY  yi | X  x 
E Y    E Y | X  xi  Pr X  xi 
i 1
k
i 1
Ea  bX  cY   a  bE( X )  cE (Y )
Var (Y | X  x)    yi  E Y | X  x  PrY  yi | X  x 
2
i 1
Cov( X , Y )   x j   X  yi  Y  PrX  x j , Y  yi 
k
m
i 1 j 1
Corr  X , Y    XY 
Cov X , Y 
Var  X Var Y 
E Y 2   Var (Y )  E (Y ) 2
1
n
n

i 1
xi
s2 
Var aX  bY   a 2Var ( X )  b 2Var (Y )  2abCov( X ,Y )
Cova  bX  cV ,Y   bCov( X ,Y )  cCov(V ,Y )
E XY   Cov( X ,Y )  E( X ) E(Y )
X 
Var a  bY   b 2Var (Y )
1 n
xi  x 2

n  1 i 1
t
X 
Z
s/ n
s XY 
For the simple linear regression model Yi   0  1 X i   i we have : ˆ1 
 X
i 1
i
 X Yi  Y 
 X
n
i 1
X
2
i

rXY  s XY / s X sY
1 n
 xi  x  yi  y 
n  1 i 1
n
X 
βˆ 0  Y  ˆ1 X
Yˆi  ˆ0  ˆ1 X 1i  ˆ2 X 2i    ˆk X ki
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