1.Text Problems: 3.1, 3.6, 3.13, 3.14, 4.1, 4.2, 4.5

2 . The Demographics Journal published the 97-98 percentage population changes
in states left (http://www.amcity.com/journals/demographics/report78/78-4.html)

Rank

1. Nevada +4.06%
2. Arizona +2.53%
3. Georgia +2.03%
3. Colorado +2.03%
5. Texas +1.93%
6. Utah +1.68%
7. Idaho +1.64%
8. Florida +1.63%
9. North Carolina +1.56%
10. California +1.51%
11. Washington +1.31%
12. South Carolina +1.26%
13. Oregon +1.19%
14. Delaware +1.15%
15. New Hampshire +1.10%
15. Tennessee +1.10%
17. Kansas +1.06%
18. Minnesota +0.81%
19. Virginia +0.80%
20. Maryland +0.78%
21. Oklahoma +0.76%
22. New Mexico +0.75%
22. Mississippi +0.75%
24. Alaska +0.71%
25. New Jersey +0.70%
26. Alabama +0.69%
27. Kentucky +0.67%
28. Arkansas +0.60%
29. Indiana +0.59%
30. Missouri +0.56%
31. Massachusetts +0.53%
32. Illinois +0.47%
33. Wisconsin +0.43%
34. Vermont +0.38%
34. Michigan +0.38%
36. Louisiana +0.35%
37. Nebraska +0.34%
38. Iowa +0.28%
39. Connecticut +0.21%
40. Montana +0.20%
41. Maine +0.19%
42. Wyoming +0.18%
43. New York +0.16%
44. Ohio +0.15%
45.Rhode Island+0.12%
46. Hawaii +0.08%
47. South Dakota +0.06%
48. Pennsylvania -0.08%
49. West Virginia -0.22%
50. North Dakota -0.42%
51. District of Columbia -1.28%

i. Construct a frequency distribution of the percentage population changes.

ii. What is the mean and the standard deviation of the percentages in the sample?

iii. Is the frequency distribution unimodal or multimodal?

iv. What possible explanation can you provide for the regional trends exhibited here?

Problem Set 2

In table 3, (see Problem set 2 for table) the data from table 1 have been converted into real terms by
dividing consumption and GDP by the price deflator and calculating a real
interest rate by subtracting the rate of increase of the deflator from the
interest rate. Table 4, which again you will use for estimating the least
squares regressions, shows the sums of squares and cross -products for the
real variables as well as their sums.

1. Calculate the simple regressions relating real consumption first to the
real GDP and second to the real interest rate. Give your numerical results
for these coefficients. Comment on any difference from the results using
the nominal values from Tables 1 and 2 and using the real values of Tables
3 and 4.

2. What dummy variable, if any, would be a logical addition for the
equation in problem 3 relating real consumption to real GDP? Supposing
there were an applicable dummy variable, how would you calculate its
coefficient?

3. Derive the expression for the simple regression
coefficient, beginning with the normal equations. Explain each mathematical
step in your own words.

Problem Set 3

1. Mirer 8.7, 13.13, Beals, 2.11, 3.1, 4.1, 4.5, 4.11

Problem Set 4

1. Problems in Text. Mirer: 9.9, 9.10,9.15, 9.20,9.23

2. The Beloit Corp., based in Beloit, Wisconsin, sells paper manufacturing
equipment. Based on past experience, we know that in the summer months it
is equally likely that it will sell 0,1,2,3, or 4 paper manufacturing
machines in a day. (The firm has never sold more than 4 machines per day.)

(a) Is the number of machines sold in a day a random variable?

(b) If so, what values can this random variable assume?

(c) If the number of machines sold in a day is a random variable, construct
a table showing its probability distribution.

(d) Plot the probability distribution of the number of machines sold in a
day in a line chart. What mathematical function can represent this probability distribution?

(e) What is the probability that the number of machines sold on a given day
will be less than 1? Less than 3? Less than 4? Less than 6?

3. An insurance company offers a 50-year-old woman a \$1,000 one-year term
insurance policy for an annual premium of \$18. If the annual number of
deaths per 1,000 is six for women in this age group, what is the expected
gain for the insurance company from a policy of this kind? What is the
standard deviation of the gain for the insurance company from a policy of
this kind?

Problem Set 5

1. Problems in Mirer: 10.1, 10.2, 10.4, 10.8, 10.9 Beals: 6.2, 6.4

2. Why do we use the standard normal distribution? How do we calculate
parameters differently on samples and populations? Explain the difference
between z-tests and t-tests.Explain the central limit theorem in your own words.

3. LSAT scores
A. Among the applicants to one law school in 1976, the average LSAT score
was around 600 with a standard deviation of 100. The LSAT scores followed
the normal curve.

i. What percentage of the applicants scored above 700?

ii. Estimate the 90th percentile for the scores.

B. Incoming students at a certain law school have an average LSAT score of
680 with a standard deviation of 25. What is your best guess for the score
of a randomly chosen student: 680, 705, or 730? Why?

4. Write your own problem which relates to z-tests, normal curves, or
sample distributions. Include a detailed solution.

Problem Set 6

1. Text problems: Mirer, 18.6, 18.7, 18.8, 18.9 Beals: 8.1a, 8.4,8.6,8.13

2. Bill has a hunch, which he decides to evaluate quantitatively. Bill
believes that the numbers of people in Springfield (total population 1996
5000) relying on state supplimentary income in the form of welfare are
''lower than they used to be.'' In order to evaluate this, he randomly
samples 1000 people in the population. He calculates a sample mean of 3
with a sample standard deviation of 1.71. A worker plotted the data and the
distribution appears to be normal. Construct a confidence interval at the
95% level of confidence for the population mean.

3. In a famous medical study, not long after the aforementioned tobacco
company investigated changing their brand variety, a researcher at Sloan
Kettering Cancer institute began studying the effects of tobacco smoke. Dr.
Wynder painted mice with a distillate of cigarette smoke, equivalent to one
pack of cigarettes three times a week. Of a randomly selected sample of 62
mice from Dr.Wynder's population, 27 mice developed cancerous tumors after
one year (Kluger 161). Construct a confidence interval for the population
proportion at the 95% level of confidence.

4. You are the fish buyer for a company that sells frozen seafood to
supermarket chains. Part of your job is to bid on boatloads of shrimp when
they arrive at the dock. The selling price of shrimp per pound depends on
the number of shrimp in a pound, large shrimp being more valuable than
shrimpy shrimp. You have to estimate the mean number of shrimp per pound
for a boatload before you know how much to bid. You randomly select 50
pounds and count the number in each pound. The sample mean is 25 with a
sample standard deviation of 6. Construct a confidence interval at the 95%
level of confidence for the mean number of shrimp per pound in the boatload.

5. According to a panel appointed by the Environmental Protection Agency,
the quantity of a much-debated ingredient in gasoline, M.T.B.E., or methyl
tertiary butyl ether, a possible carcinogen, should be ''reduced
substantially'' because it dissolves easily in water and turns up in tap
water when gasoline has leaked or spilled. MTBE was introduced into
gasoline, in order to meet 1990 federal clean air regulations; experts
differ on how much the ingredient has done to control carbon monoxide and
smog, but they agree that it has cut toxic components from gasoline in the
air by about 30 percent (7/27/99 NYT Wald). A major gasoline manufacturer
would like to demonstrate that M.T.B.E has significantly reduced the parts
per million of carbon monoxide in the air. Average ppm of carbon monoxide
in the air in Milwaukee, Wisconsin before MTBE was 346. Consumer advocates
observe that on 16 days when air quality was measured during the year after
the MTBE was introduced there was an average reading of 341 ppm with a
sample standard deviation of 36. Test at the 5% significance level the
hypothesis that the mean reading is equal to 346, assuming a normal
distribution.

Problem Set 7

1. Text Problems: Mirer 11.10, 11.11, 11.16,11.18, 12.3, 12.5, Beals 10.3

Problem Set 8

1. Prove [ (R^2)/[k-1]/{(1-R^2)/(N-k)}=[{(ESS)/[k-1]}/{(RSS)/(N-k)}

2. Demonstrate: for the addition of one variable, the F statistic=the
square of the t statistic.

3. Suppose that you wanted to study the behavior of cigarette smokers over
a number of years, and the statisticians at the Department of Health and
Human Services suggest two models for cigarette consumption:
Y_t=B₀+B₁t and Y_t=a₀+a₁t+a₂t² where

Y_t =consumption at time t, t=time measured in years.

a) What pattern of cigarette consumption does the first model suggest? The
second?

b) How would you choose between the two models?

c) For what other products might the quadratic be applicable?

4. Suggest another macro model, and write out your null and alternative
hypotheses for two different f-tests.

Problem Set 9

1. Text problems: Mirer 6.20, 6.22, 6.24, 6.26, Beals 12.1, 12.3,

b) Let Y be the price of a house and X be the living area. We can expect
that the additional price a consumer will be willing to pay for 100
additional square feet of living area will be higher when X=1300 than when
X=3200. This means that the marginal effect of SQFT on Price can be
expected to decrease as SQFT increases.

i)What form would enable us to capture this relationship?

ii) Suggest what equation you would use to find the elasticity at the mean
price of house.

2. Demand functions are frequently written as reciprocal transformations

Y= B₀+B₁(1/X).

a)What sign would we expect B₁ to have?
b)What value does Y approach as X goes to infinite?
c)Graph this line for B₁ <0 and B₁ >0.
d)Suggest how you might calculate the elasticity of this demand function.

3. Quadratic functions are frequently used to fit u-shaped cost functions.
For the given quadratic function, of the form Y=B₀+B₁X+B₂X²+e_t we have data on the per-unit cost of a manufacturing firm over a 20-year
period, an index of its output, and an index of its input costs. We have
estimated a regression equation (with t-values in parentheses).

UnitCost=	10.522	-.175 Output	+.00895 Output2	+.0202 Impcost
	(14.3)	(-9.7)	(7.8)	(14.454)
R2=.978	d.f.=16

a) Test the significance of each of the variables.

b) Sketch what you think this estimated average cost function would look like.

c) For an input cost of 80, plot a few sample points using output values of
60, 80, 100, and 120 (Ramanathan).

d) Find another function, in your Micro book which follows this form, and
suggest likely variables.

4. Suppose that, instead of the linear demand for money function in
Exercise 1, you hypothesize the log-log demand function

ln M_t=a₀+a₁lnY_t+a₂lnI_t+e_t

a) Find the least squares estimates of a₀,a₁, and a₂.
b) Interpret the estimated coefficients.

c) Using the results you obtained in Exercise 1 and 2a, find 95% interval
estimates for B₁, B₂,a₁, and a₂.

d) Test each of the null hypotheses H₀: B₁=B₂ =0 and
H₀:a₁= a₂=0.

e) Is there any evidence in the data to suggest that the linear or the
log-linear model is more appropriate?

Problem Set 10

1. You are considering betting $100 on a horse named Bayes. If the horse
wins, you will get your $100 back and an additional $900. If it loses, you
will be out the $100. Your subjective probability of the horse's winning is
0.2. You could buy a tip from Mr. Oracle before making the bet. Given that
a horse will lose, the probability that Mr. Oracle will predict that it
will win is 0.05. Given that a horse will win, the probability that Mr.
Oracle will predict a win is 0.5. The probabilities of Mr. Oracle's
prediction that Bayes will win or lose are 0.1 and 0.9 respectively. Mr.
Oracle's prediction cost $100. Construct a tree diagram for this problem
and decide on the best course of action. What is the expected payoff of
this best choice?

2.An appliance firm must decide whether or not to offer Mr.Jones a job. If
Mr.Jones is a success, the firm will increase its profits by 100,000; if he
turns out not to be a success, the firm's profits will decrease by 80,000.
The firm feels the chances are 50-50 that he will be a success (courtesy of
Mansfield 17.7).

A) How can the firm obtain information concerning whether
or not Whelan will be a success?

B) What is the expected value of Mr.Jones
to the company? Should they hire him?

3. In the game Rock, Scissors, paper, two players simultaneously show hand
signals representing a rock, scissors, or paper. Rock beats scissors,
scissors beats paper, and paper beats rock. Show the payoff matrix using 1
to represent a win, -1 to represent a loss, and 0 to represent a tie. Is
there a noncooperative equilibrium? What mixed strategy would you
recommend? (Smith 19.32,33)

4. One player plays rock 26 times, scissors 9 times, and paper 15 times.
This is your opponent; these empirical frequencies are the probabilities
that this student's first move will be rock, scissors, or paper (Smith
19.58). What strategy mazimizes the expected value of your playoff?

5. Bobby M. drives to Wharton every day. He can either turn right, from
the highway, into parking lot P where there is always plenty of space or he
can take his chances, driving ten minutes farther, into the small parking
lot behind the Business school (30 percent chance of finding a spot). If he
does not find a parking space in the small parking lot, he can always park
at the meter in front of the B School. If he's feeling punchy, he can forgo
the daily four dollar meter fee, and instead chance a 15 dollar ticket
(occurs one tenth of the time). It takes 10 minutes to walk to the B school
from parking lot P, two minutes from the small parking lot, and 4 minutes
from the meter at front (Wonnacott and Wonnacott, 21.11). If his time is
worth 12 per hour, what is his best strategy? How much does it cost?
Suppose he faces an additional penalty for not paying the meter: Half the
cars that get a ticket also get towed away at a cost of 40, on top of the
15 ticket, plus an hour of time. Now, what should he do? At what cost?

 Problem Set 11

1. Suppose that a researcher is trying to estimate the following
consumption function, where consumption is hypothesized to depend on
current income (Y_t) habit or past income (Y_t-1), and
expectations or change in income deltaY=(Y_t-Y_t-1)(JJB):
E_t=b₀+ b₁ Y_t +b₂Y_t-1+b₃(deltaY)+e_t.
Show that without further information it is impossible to estimate
b₁,b₂, and b₃.

2. Assume that the following production function is to be estimated from
annual data on a firm (JJB):
Q=B₀ + B₁L₁+B₂L₂+B₃K

L₁= acres of land
L₂= dollars of labor
K= dollars of capital equipment

Suppose that the firm always budgets $30,000 a year for labor and capital

(i.e. L₂+K=30,000)

a) Is there a multicolinearity problem?

b) Can the coefficients be estimated? 

3. (Studenmund). Which of the following pairs of variables is likely to
include a ''redundant'' variable?

A) the price of refridgerators and the price of washing machines in a
durable-goods demand function

B) The number of acres harvested and the amount of seed used in an
agricultural supply function

C) long-term interest rates and the money supply in an investment function

 4. The following regression model is heteroscetastic: (1) Y=B₀+B₁X₁+B₂X₂+U where Y=dollars spent by consumers

on food in a city in a week, X₁= dollars of income earned by consumers
in a city during a week, and X₂=population of a city (JJB). A
researcher uses OLS to estimate (1) by the following equation, assuming
that E(U²)=sigma² . Y=b₀+b₁X₁+b₂X₂+e

a) Is E(b₀)=B₀?Is E(b₁)=B₁?Is E(b₂)=B₂? E(S²)=sigma²?
b) Assuming that b₀ ,b₁ and b₂ are unbiased estimators of the
parameters in 1, can S_bi be used to test hypothesis?

5. Autocorr/heterosketasticity (JJB).

A) Which type of mispecified
disturbance term is more likely to occur in time-series data? In cross
sectional data?

B)Is the variance of the disturbance term more likely to be positively or
negatively correlated with the independent variables?

C)Is autocorrelation more likely to be positive or negative?