SAMPLE MULTIPLE CHOICE QUESTIONS FOR MIDTERM
1.) Suppose the monthly demand for tomatoes (a perishable good) in a small town is random. With
probability 1/2, demand is 50; with probability 1/2, demand is 100. You are the only producer of tomatoes
in this town. Tomatoes sell for a fixed price of $1, cost $0.50 to produce, and can only be sold in the
local market. If you produce 60 tomatoes, your expected profit is:
a) $15
b) $25
c) $45
d) $50
e) none of the above
2.) Suppose you have the following information about the cdf of a random variable X, which takes one of
4 possible values:
Value of X
Cdf
1
0.25
2
0.4
3
0.8
Which of the following is/are true?
a) Pr(X = 2) = 0.4
b) E(X) = 2.5
c) Pr(X = 4) = 0.2
d) all of the above
e) none of the above
3.) 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
4.) If two random variables X and Y are independent,
a) their joint distribution equals the product of their marginal distributions
b) the conditional distribution of X given Y equals the marginal distribution of X
c) their covariance is zero
d) a and c
e) a, b, and c
4
5.) Suppose you have a random sample of 10 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 xbar 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 2
e) none of the above
6.) If q is an unbiased estimator of Q, then:
a) Q is the mean of the sampling distribution of q
b) q is the mean of the sampling distribution of Q
c) Var[q] = Var[Q] / n where n = the sample size
d) q = Q
e) a and c
7.) Suppose you compute a sample statistic q to estimate a population quantity Q. Which of the following
is/are false?
[1] the variance of Q is zero
[2] if q is an unbiased estimator of Q, then q = Q
[3] if q is an unbiased estimator of Q, then q is the mean of the sampling distribution of Q
[4] a 95% confidence interval for q contains Q with 95% probability
a) 2 only
b) 3 only
c) 2 and 3
d) 2, 3, and 4
e) 1, 2, 3, and 4
8.) 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
9.) Suppose you draw a random sample of n observations, X1, X2, …, Xn, from a population with unknown
mean μ. Which of the following estimators of μ is/are biased?
a) the first observation you sample, X1
b)
X2
c) X 2  s 2 / n
d) b and c
e) a, b, and c
10.) The significance level 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
11.) Suppose you want to test the following hypothesis at the 5% level of significance:
H0: μ = μ0
H1: μ ≠ μ0
Which of the following is/are true?
a) the probability of a Type I error is 0.05
b) the probability of a Type I error is 0.025
c) the t statistic for this test has a t distribution with n-1 degrees of freedom
d) a and c
e) b and c
12.) Suppose [L(X), U(X)] is a 95% confidence interval for a population mean. Which of the following
is/are true?
a) Pr L X   X  U  X   0.90


b) PrL X   X  U  X   0.95
c) PrX  L X  PrU  X   X   0.05
d) a and c
e) none of the above
13.) Which of the following is a linear regression model:
a) Yi    1 X i   2 X i2   i
b) log( Yi )  0  1 log( X i )   i
c) Yi  0  1e X i   i
d) all of the above
e) none of the above
14.) In the linear regression model, the stochastic error term:
a) measures the difference between the dependent variable and its predicted value
b) measures the difference between the independent variable and its predicted value
c) is unbiased
d) a and c
e) none of the above
15.) 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
16.) Suppose that in the simple linear regression model Yi = β0 + β1Xi + εi on 100 observations, you
calculate that R2= 0.5, the sample covariance of X and Y is 10, and the sample variance of X is 15. Then
the least squares estimator of β1 is:
a) not calculable using the information given
b) 1/3
c) 1 / 3
d) 2/3
e) none of the above
17.) Suppose upon running a regression, EViews reports a value of the explained sum of squares as 1648
and an R2 of 0.80. What is the value of the residual sum of squares in this case?
a.) 0
b) 412
c) 1318.4
d) unknown as it is incalculable
e) none of the above
18.) In the linear regression model, adjusted R2 measures
a) the proportion of variation in Y explained by X
b) the proportion of variation in X explained by Y
c) the proportion of variation in Y explained by X, adjusted for the number of independent
variables
d) the proportion of variation in X explained by Y, adjusted for the number of independent
variables
e) none of the above
19.) In the linear regression model, the degrees of freedom
a) is equal to the number of observations (n) minus 1
b) affects the precision of the coefficient estimates
c) affects the value of the coefficient estimates
d) all of the above
e) none of the above
20.) In the Capital Asset Pricing Model (CAPM),
a) β measures the sensitivity of the expected return of a portfolio to systematic risk
b) β measures the sensitivity of the expected return of a portfolio to specific risk
c) β is greater than one
d) α is less than zero
e) R2 is meaningless
1.) Suppose the monthly demand (x) for a perishable good is a random variable that takes one of six
possible values. The pdf of monthly demand is f(x):
x
f(x)
100
0.10
200
0.10
300
0.20
400
0.35
500
0.20
600
0.05
The good sells for a fixed price of $15 per unit, and production costs are $10 per unit. Therefore, the firm
earns $5 profit on each unit sold and loses $10 on each unit that goes unsold. If the firm brings 400 units
to market, what is the expected profit? What is the variance of profit?
2.) Suppose the price of a stock X is a random variable. On any day, its value may increase, decrease, or
not change at all. The distribution of daily price changes is as follows:
Price Change ($)
Probability
-1.00
0.283
0.00
0.25
0.50
1.00
0.10
a) What is the probability of a $0.5 increase in price?
b) Draw the pdf and cdf of price changes.
c) What is the expected price change?
d) What is the variance of the price changes?
e) Suppose the stock’s price today is $10. What is the expected value of tomorrow’s price?
What is its variance?
Now suppose the distribution above only applies when the weather is sunny. When the weather is rainy,
the distribution is:
Price Change ($)
Probability
-1.00
0.50
0.00
0.20
0.50
0.20
1.00
0.10
f) What is the expected price change on a rainy day?
g) Suppose the probability of rain is 0.4, and the probability of sun is 0.6. What is the
expected price change?
3.) Suppose you collect the following data that are a random sample from a N(μ,σ2) population:
4.37 6.99 7.85 2.60 3.34 5.94 4.21 5.99 8.53 4.92
a) Compute the t-statistic for testing the hypothesis:
H0 : μ = 4
H1 : μ ≠ 4
b) What is the sampling distribution of the test statistic you computed in part a)?
c) Can you reject the null hypothesis of part a at the 5% level of significance? Explain.
4.) Suppose you have a random sample of 100 SFU students. In response to a short survey, each student
reported the usual number of hours per week that he/she spent working off-campus (Xi), and the usual
number of hours per week that he/she spent engaged in social activities (Yi). The university has hired you
to analyze these data. Their main interest is the total number of hours per week that SFU students engage
in non-academic activities. Define a new variable, Zi = Xi + Yi, to measure the number of hours that SFU
students engage in non-academic activities. Suppose that in the population of SFU students:
EX i    X , EYi   Y ,Var X i    X2 ,Var Yi    Y2 , CovX i , Yi    XY .
a) The university has asked you to estimate the average number of hours that SFU students
engage in non-academic activities, i.e., E[Zi] = μz. Your roommate, who has already taken BUEC
333, says “That’s easy! Just take the first person in your sample. Their value, Z1, is an unbiased
estimator of μz.” Is your roommate right? Explain.
b) What is the variance of your roommate’s estimator in part a)?
c) Give a more efficient, but still unbiased, estimator of μz. Show that it is unbiased and that it
is more efficient than your roommate’s estimator.
d) What is the variance of the sampling distribution of your estimator in part c)? Explain
what the variance of the sampling distribution measures.
5.) Suppose we have a linear regression model with one independent variable and no intercept:
Yi = βXi + εi
a) Verbally explain the steps necessary to derive the least squares estimator (hint: this should entail
four distinct steps).
b) Formally derive an expression for this estimator given your answer in part a).