Download Midterm 2002 key

Oct. 31, 2002
Midterm Key
Econ 240A-1
L. Phillips
Answer all five queations.
1. (15 points) An investor believes that on a day when the Dow Jones Industrial Average
(DJIA) increases, the probability that the NASDAQ also increases is 77%. If the
investor believes that there is a 60% probability that the DJIA will increase tomorrow,
what is the probability that the NASDAQ will increase as well?
Answer: Prob(NASDAQ up/DJIA up) = 0.77
Prob(DJIA up tomorrow) = 0.60
Prob (NASDAQ up) = Prob(NASDAQ up/DJIA up)* Prob(DJIA up tomorrow)
= 0.77*0.60 = 0.462
2. (15 points) The boss assigns you to choose an internet service provider (ISP). You
want an ISP that is large enough so that a customer (caller) seldom receives a busy
signal. With the ISP you choose, a customer (caller) encounters a busy signal only 8%
of the time. Your office makes 60 calls per week.
a. What is the probability that callers from your office do not encounter any busy
signals in a week? Note: Full credit for the correct numerical expression for
the probability. You do not have to calculate the numerical value of this
expression in part a.
Answer: each call is a Bernoulli event with a success being no busy signal with
probability 0.92 and a failure being a busy signal with probability 0.08. Using the
binomial distribution, the probability of 60 successes in 60 calls is:
P(k=60) = 60!/(60!*0!)(0.92)60(0.08)0 = 1*)(0.92)60*1 = (0.92)60
This is a satisfactory answer for this exam. LnP(k=60) = 60*ln(0.92) =-5.0028965
So exp(5.002895 = 0.0067 = Prob(k=60)  0.01
b. Suppose you want to approximate an answer for this probability. Provide a
numerical approximate value for the probability that callers from your office
do not encounter a busy signal in a given week.
Answer: Using the normal approximation, the E(k) = n*p = 60*0.92 = 55.2.
Oct. 31, 2002
Midterm Key
Econ 240A-2
L. Phillips
The VAR(k) = n*p*(1-p) = 60*(0.92)*(0.08) = 4.416. The standard deviation is
the square root of 4.41 which equals 2.1. The probability of getting less than 60
successes would be the probability of falling below a standardized z of:
Z = (60-55.2)/2.1 = 2.29. From Table 3 in Appendix B, the area in the upper tail
of the normal distribution for z = 2.29 is 0.011, so the chances that no one in your
office gets a busy signal are pretty low.
c. Is this approximation pretty good? Explain.
Answer: yes the criteria for the approximation are n*p  5 and n*(1-p)  5 . The
first is 60*0.92 and is clearly satisfied, the second is 60*0.08 = 4.8 and is close.
3. (15 points) Americans tend to eat too much fast food. A doctor claims that the
average Californian is more than 20 pounds overweight. To check his claim, you take
a random sample of 20 Californians and weigh them. The difference between their
observed weight and their ideal weight follows: 16, 23, 18, 41, 22, 18, 23, 19, 22, 15,
18, 35, 16, 15, 17, 19, 23, 15, 16, 26. The mean of this sample is 20.85, and the
standard deviation of this sample is 6.76.
a. From this sample data, do you think the doctor's claim is true?
Answer: The null hypothesis is that the population mean is equal to 20, and the
alternative hypothesis is that the population mean is >20. Since the population
mean is not known, use Student’s t-distribution. The t-statistic is:
t = [ x  E ( x )] / s n = [20.85 – 20]/(6.7 20 ) = 0.85/(6.76/4.47) =
0.85/1.51=0.56, where E( x ) = the population mean under the null hypothesis, i.e.
20. So do not reject the null. For 19 degrees of freedom, i.e n-1, at the 5% level
the critical t-statistic is 1.73, so the calculated t-statistic is well below the critical
t. for a 5% type I error.
4. (15 points) The following graph plots the UC budget, the component funded by the
state, against California Personal Income, both in billions of nominal dollars. A linear
trendline has been fitted to the data.
a. How would you describe the goodness of fit?
Oct. 31, 2002
Midterm Key
Econ 240A-3
L. Phillips
Answer: It is fairly good in that R2 is 95%, but there are some sizeable errors in
the 80’s and 90’s, before and after the recession of 1991.
b. Would you use only this trendline to predict next year's UC budget?
Answer: It would be a good idea to calculate several forecasts, for example one
from problem 5 below, as well.
c. From the information provided about past experience, if California Personal
Income goes up by a 100 billion next year, how much would you expect the
UC budget to increase?
Answer: From the slope of 0.0027, about 0.27 billion or 270 million
UC Budget, General Fund Component Vs. CA Personal Income, Both in Billions of Nominal $,
1968-69 through 2002-2003
4
3.5
y = 0.0027x + 0.1972
R2 = 0.9513
3
UC Budget
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
CA Personal Income
Figure 4.1 UC Budget, General Fund Component Vs. CA Personal Income, both
in Billions of Nominal Dollars, 1968-69 through 2002-2003
5. (15 points) In the figure below, the UC Budget, General Fund Component, is plotted
against California Personal Income, both in billions of nominal dollars, from fiscal
year 1968-69 through 2002-03, on a log-log scale. In the table that follows, the results
Oct. 31, 2002
Midterm Key
Econ 240A-4
L. Phillips
of regressing the natural logarithm of the UC Budget against the natural logarithm of
California Personal Income follows.
a. Is this regression statistically significant? Explain.
Answer: yes, the F-statistic of 1598 is highly significant.
b. Interpret the estimated slope coefficient.
Answer: it is the elasticity of the UC Budget to CA personal Income, indicating
for every 10% increase in CA personal Income, the UC Budget goes up 8.78%.
c. Is this slope coefficient significantly different from zero?
Answer: Yes, the t-statistic from Table 5.1 of 40 is highly significant.
d. Is this slope significantly less than one?
Answer: the null hypothesis is that the slope is one and the alternative hypothesis
is that the slope is less than one. The t-statistic is:
[ ˆ  E ( ˆ ) /  ( ˆ ) ] = (0.878-1.0)/0.0219 = -0.122/0.0219 = -5.57 where the
expected value of the estimated slope is 1 under the null hypothesis. So the
elasticity is significantly less than one.From Table 4 in Appendix B, for 33 = n-2
degrees of freedom, the critical t-statistic is –1.70, since the t-distribution is
symmetric.
e. Which model appears to fit better, the linear one or the proportional(power
function) one?
Answer: the proportional has a higher R2, but they are comparable.
UC Budget, General Fund Component Vs. CA Personal Income, both in Billions of Nominal $
UC Budget
10
y = 0.0067x0.8779
R2 = 0.9798
1
10
100
1000
0.1
CA Personal Income
10000
Oct. 31, 2002
Midterm Key
Econ 240A-5
L. Phillips
Figure 5.1 UC Budget, General Fund Component Vs. CA Personal Income, both in
Billions of Nominal $, 1968-69 through 2002-03, log-log scale
--------------------------------------------------------------------------------------------------------Table 5.1 Regression of the Natural Logarithm of the UC Budget, General Fund
Component, on the Natural Logarithm of CA Personal Income, both in Billions of
Nominal $, 1968-69 through 2002-03
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.98983424
R Square
0.97977182
Adjusted R Square 0.97915884
Standard Error
0.10679464
Observations
35
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
33
34
SS
MS
F
Significance F
18.22975615 18.22976 1598.387 1.55548E-29
0.376368113 0.011405
18.60612426
Coefficients Standard Error
t Stat
P-value
Lower 95%
-5.00664713
0.131107382 -38.1874 6.85E-29 -5.273387312
0.87791675
0.021958989 39.97983 1.56E-29 0.833240815
Upper 95%
Lower 95.0% Upper 95.0%
-4.73990694 -5.273387312 -4.73990694
0.922592685
0.833240815 0.922592685