HW 1 solutions

Econ 526 / Fall 2014
Manopimoke
Econometrics Problem Set #1
SOLUTIONS
Instructions:
Please write your answers legibly. No late assignments, no exceptions.
1. Faced with the question of determining the probability of obtaining either 0 heads, 1 head or 2
heads when flipping a coin twice, an individual argued that since it seems reasonable to treat
each outcome as equally likely, the probability of each event is 1/3. Do you agree or disagree. If
you disagree, what are the actual probabilities to the above outcomes? (5 points)
Answer this question by finding the probability of each outcome. You have the following
sample space. {(H,H) (H,T) (T,H) (T,T)}. The P(no heads) = 1/4. P(1 head) = 1/2. P(2
Heads) = 1/4. Thus, the probability of each outcome is not 1/3.
2. A deck of cards is shuffled and three cards are dealt. What is the chance that the first card is a
king, the second card is a king, and the third card is a jack? (5 points)
This is a conditional probability. P(K) * P(K|K) * P(J|K, K) = 4/52 * 3/51 * 4/50 =
48/132,600 = .0004.
3. On a multiple choice test with 50 questions, each question has 4 possible outcomes (a, b, c, d)
one of which is correct. For students who guess at all of the answers, find the mean, variance
and standard deviation for the number of correct answers. (10 points)
Use the formula for the mean and variance of the binomial distribution. N == 50 and p =
.25. Then the mean for guessing at all answers: µ=50*.25 = 12.5. The variance:
2 = n * p **q = 50*.25*.75 = 9.375 and the standard deviation: = n * p * q = 3.062.
4. A statistics professor gave the class two different tests designed to measure the understanding
of elementary statistics. The student can choose which test will count towards the final grade.
Would the student rather have a score of 57 on the first test where the mean is 72 and the
standard deviation is 20 or a score of 450 on the second test where the mean is 500 and the
standard deviation is 80. Please explain. (15 points)
Use Z-scores to compare two distributions.
x - x 57 - 72
=
= - .75
20
s
x - x 450 - 500
=
= - .63
z1 =
80
s
A student would rather have a score closer to the mean, thus, they would choose to have the
test with a score of 450 count towards their grade.
z1 =
5. Assume that women have heights that are normally distributed with a mean of 63.6 inches
and a standard deviation of 2.5 inches. Find the probability that a woman is (15 points):
(A) Under 66 inches.
P( x < 66) =
= P(z<.96) = .5 + F(.96)
= .5 + .3315 = .8315
(B) Over 72 inches.
P( x > 72) =
= P(z > 3.36) = .5 - .4999 = .0001
(C) Between 66 and 72 inches.
P( 66 < x < 72) = P(.96 < z < 3.36)
= F(3.36) - F(.96) = .4999 - .3315 = .1684
6. A New York Times article noted that the mean life span for 35 male symphony conductors
was 73.4 years, in contrast to the mean of 69.5 years in the general population. Assuming these
35 conductors have a standard deviation of life spans equal to 8.7 years, test the claim that
symphony conductors mean life span is different than men in the entire population. Use a .05
level of significance (10 points)
H0 : µ = 69.5
H A : µ ≠ 69.5
2) This is a two-tailed test. We use the z-score to test this hypothesis:
1)
z=
X −µ
s
n
73.4 − 69.5
8.7
35
z = 2.65 > 1.96 z (critical at .05).
z=
7. A random sample of 250 credit card holders shows the mean annual credit card debt for
individual accounts is $1592, with a standard deviation of $997. Use these statistics to construct
a 94% confidence interval for the mean annual credit card debt for the population of all accounts.
(10 points)
Creating a 94% confidence interval requires finding the critical value. 94/2 = .47.
Using the z table .47 corresponds to a z score of 1.88.
Now calculate the confidence interval:
= [1473.44, 1710.56]
8. Macroeconomic Data Analysis: Go to the Federal Reserve Bank of St. Louis data website,
FRED http://research.stlouisfed.org/fred2/. Download quarterly GDP from 1970 and the
quarterly GDP implicit price deflator. Put these data into a spreadsheet and convert the base
year of the deflator to 1990. Create the real GDP series. Graph the nominal and real GDP
series. When does nominal GDP equal real GDP? Why? Please attach your graphs and
interpretation to the problem set that you hand in.
Nominal GDP equals Real GDP in 1990. This is because Real GDP = Nominal GDP/ GDP
deflator and the GDP deflator is equal to 1 in the base year.
Nominal GDP
Real GDP Base 1990
2010-01-01
2008-05-01
2006-09-01
2005-01-01
2003-05-01
2001-09-01
2000-01-01
1998-05-01
1996-09-01
1995-01-01
1993-05-01
1991-09-01
1990-01-01
1988-05-01
1986-09-01
1985-01-01
1983-05-01
1981-09-01
1980-01-01
1978-05-01
1976-09-01
1975-01-01
1973-05-01
1971-09-01
1970-01-01
Real and Nominal GDP Growth 1970-2011
16000.0
14000.0
12000.0
10000.0
8000.0
6000.0
4000.0
2000.0
0.0

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