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Practice for Exam 1 Econ
B2000, MA Econometrics Kevin R Foster, CCNY Fall
2013 |
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Not all of these questions are strictly relevant; some might require
a bit of knowledge that we haven't covered this year, but they're a generally
good guide.
1.
(15 points)
i.
For a Standard Normal Distribution, what is area to the left of
2.1? A. 0.5328 B. 0.0357 C. 0.3507 D. 0.9821
ii.
For a Standard Normal Distribution, what is area to the left of
-0.6? A. 0.2743 B. 0.0547 C. 0.1368 D. 0.2002
iii.
For a Standard Normal Distribution, what is area to the left of
0.3? A. 0.3821 B. 0.9390 C. 0.6179 D. 0.7642
iv.
For a Standard Normal Distribution, what is area to the right of
0.9? A. 0.1841 B. 0.0875 C. 0.3684 D. 0.0552
v.
For a Standard Normal Distribution, what is area to the right of
-0.4? A. 0.5870 B. 0.3446 C. 0.6225 D. 0.6554
vi.
For a Standard Normal Distribution, what is area in both tails
farther from the mean than 1.8? A. 0.9641 B. 0.1841 C. 0.1437 D. 0.0719
vii.
For a Standard Normal Distribution, what is area in both tails
farther from the mean than 0.5? A. 0.0760 B. 0.5987 C. 0.6915 D. 0.6171
viii.
For a Standard Normal Distribution, what is area in both tails
farther from the mean than -2.4? A. 0.9918 B. 0.0328 C. 0.0164 D. 0.0470
ix.
For a Standard Normal Distribution, what values leave
probability 0.324 in both tails? A. ±0.4565 B. ±1.2305 C. ±0.9863 D. ±0.3799
x.
For a Standard Normal Distribution, what values leave
probability 0.390 in both tails? A. ±0.8596 B. ±0.8596 C. ±1.1948 D. ±0.4298
xi.
For a Standard Normal Distribution, what values leave probability
0.218 in both tails? A. ±0.7790 B. ±0.1611 C. ±1.2319 D. ±1.2957
2.
(15 points)
i.
For a Normal
Distribution with mean 12 and standard deviation 8.9, what is area to the left
of 4.9? A. 0.0888 B. 0.4237 C. 0.2125 D. 0.7881
ii.
For a Normal
Distribution with mean -2 and standard deviation 8.3, what is area to the left
of -19.4? A. 0.9945 B. 0.0357 C. 0.8531 D. 0.0179
iii.
For a Normal
Distribution with mean
3 and standard deviation 2.6, what is area to the right of 4.3?
A. 0.1243 B. 0.3085 C. 0.6171 D. 0.6915
iv.
For a Normal
Distribution with mean
1 and standard deviation 2.3, what is area to the right of 0.8?
A. 0.1587 B. 0.5346 C. 0.3319 D. 0.5199
v.
For a Normal
Distribution with mean
5 and standard deviation 5.4, what is area in both tails farther
from the mean than 10.4? A. 0.1587 B. 0.6346 C. 0.8413 D. 0.3173
vi.
For a Normal
Distribution with mean -7 and standard deviation 2.9, what is area in both
tails farther from the mean than -5.3? A. 0.3821 B. 0.1587 C. 0.7257 D. 0.5577
vii.
For a Normal
Distribution with mean 11 and standard deviation 8.2, what is area in both
tails farther from the mean than 1.2? A. 0.1587 B. 0.2320 C. 0.8849 D. 0.7257
viii.
For a Normal
Distribution with mean
8 and standard deviation 4.2, what values leave probability 0.046
in both tails? A. (-4.3807, 12.3807) B. (-0.0906, 3.9002) C. (-0.3807, 16.3807)
D. (2.4201, 13.5799)
ix.
For a Normal
Distribution with mean -6 and standard deviation 6.7, what values leave
probability 0.347 in both tails? A. (-9.1504, -2.8496) B. (-1.8359, 0.0449) C.
(-8.6360, -3.3640) D. (-12.3008, 0.3008)
x.
For a Normal
Distribution with mean
5 and standard deviation 0.1, what values leave probability 0.301
in both tails? A. (4.9478, 5.0522) B. (4.8966, 5.1034) C. (4.9657, 5.0343) D.
(-1.0343, 1.0343)
3.
(15 points) A
(joking) study in the New England Journal of Medicine linked a country's per
capita consumption of chocolate with the number of Nobel Prizes. It reported a regression coefficient but I
got the data and did my own analysis.
Five countries with the highest consumption of chocolate (UK, Belgium,
Germany, Norway, Switzerland) had 19.02 Nobel Prizes per 10m people (std dev
9.0); the next five countries (USA, Finland, Denmark, Austria, Ireland) with
lower chocolate consumption had 16.13 prizes (std dev 8.1).
a.
Construct a
hypothesis test for whether there is no statistically significant difference between the most
chocolate-consuming countries and the next group. What is the standard error of the difference? What is the test statistic? What is the p-value?
b.
Discuss the
study. The countries that ate the most
chocolate consumed 9.28 kg/person (std
dev 0.54) while the next group ate merely 7.34
kg/person (std dev
1.52). Should you have eaten a chocolate
bar before this exam?
F
H Messerli (2012). "Chocolate Consumption,
Cognitive Function, and Nobel Laureates" N Engl J Med 367: 1562-1564 October 18,
2012
4.
(15 points) A (not joking) report from Morgan Stanley
reported that the bank's positions were 23% safer because the bank chose to
measure risk with the standard deviation of stock returns from the past one
year rather than more years as it had done previously. Over the past one year, the average daily
returns on the S&P500 (expressed at monthly rate) were 1.7% with a standard
deviation of 3.6% (252 observations).
For the four years before, the average return was -0.6% with a standard
deviation of 6.5% (1008 observations).
Test the null hypothesis that the returns for the past year are the same
as the returns for the previous four years.
What is the standard error of the difference? What is the test statistic? What is the p-value? Discuss. Why might Morgan Stanley have chosen
that particular data? (Note that
riskiness is a cost so reduces profits.)
The Excel file with this data is on
Blackboard although you do not need to use it.
5.
(15 points) Using the PUMS data in SPSS (available from Blackboard), consider
the fraction of people who are elderly (age greater than 65). Compare Bronx and Brooklyn to determine if
there is a statistically significant difference in the fraction of the
population who are elderly in these two boroughs. What is the p-value for the
null hypothesis of no difference? Which
borough has the fewest elderly people?
Which has the most? Can you find
sub-groups that explain some of the difference in averages between
boroughs? What if you break it down
further by neighborhood (PUMA)?
6.
(15 points) You might find it useful to sketch
the distributions.
a.
If a variable has
a Standard Normal Distribution, what is the probability of observing a value
less than 0.3?
b.
If a variable has
a Standard Normal Distribution, what is the probability of observing a value
farther from the mean (both tails) than 1.8?
c.
If a variable has
a Normal Distribution with mean –4 and standard deviation 7.4, what is the probability of
observing a value less than –7.7?
d.
If a variable has
a Normal Distribution with mean 6 and standard deviation 5, what is the probability of
observing a value greater than 7?
e.
For a Normal
Distribution with mean 3 and standard deviation 9.8, what values leave
probability 0.02 in both tails?
7.
(15 points) A
survey from eFinancialCareers found that, despite
predictions from NY State that the Wall St bonus pool
would drop by about 35%, a full 48% of the 911 respondents believed that their
own bonuses would rise.
a.
Test the null
hypothesis that the fraction of respondents expecting a bigger bonus is
different from 35%. What is the p-value?
b.
Create a 95%
confidence interval for the fraction expecting a bigger bonus. What is the 90% confidence interval? The 99% interval?
c.
Discuss. What other survey question might help explain
this difference?
8.
(15 points) Dan Ariely
and co-authors report a study that asks participants to solve complicated
addition tasks but gives them an opportunity to cheat: they self-report how
many problems they correctly solve.
Every participant got a pair of fashion sunglasses but some were told
that the sunglasses were counterfeit.
Forty-two people were told they got counterfeit sunglasses and 30 of
them cheated; 43 people were told that they got authentic sunglasses and 13 of
them cheated.
a.
Test the null
hypothesis that there is no difference in the rates of cheating for people
wearing counterfeit or authentic sunglasses. What is the p-value?
b.
Create a 95%
confidence interval for the difference in cheating rates. What is the 90% confidence interval? The 99% interval?
c.
Discuss. Are there other factors that could be at
work?
F
Gino, M I Norton, D Ariely (2010). "The Counterfeit Self: the Deceptive
Costs of Faking It," Psychological
Science 21:712.
9.
(15 points) An audit study emailed professors
to ask for an appointment but the names of the 'students' were randomly varied
to be typically male or female; white, African-American, Hispanic, Chinese, or
Indian. White men were 26% more likely
to get an appointment than minority women.
Suppose you wanted to do a replication study for CUNY faculty. The original study emailed 6500 professors,
you would like to study fewer.
a.
If the true
difference in response to white male vs other names
is 26%, what is the minimum sample size that could distinguish a statistically
significant difference (at 95% level)?
b.
If the true
difference were half as large, what is the minimum sample size to distinguish a
difference?
c.
What if the true
difference were just 6%? What is the
minimum sample size necessary?
d.
Discuss. Can you suggest additional tests?
K L Mikman, M Akinola, D Chugh, 2012.
"Temporal Distance and Discrimination: An Audit Study in
Academia," Psychological Science 23:7.
10.
(15 points) In
recent news a study of adolescent girls compared those who had received a
vaccination against HPV (a sexually transmitted virus that is linked to certain
cancers) with those who had not received the vaccine. Some parents had been reluctant to get their
children vaccinated because they believed this would encourage sexual
activity. The study compared 493 who got
the vaccine agains 905 who did not. Of the girls who got the vaccine, 61 got any
of testing, diagnosis or counseling for pregnancy/sexually-transmitted disease;
of those who did not get the vaccine, 76 got testing, diagnosis, or counseling.
a.
Test the null
hypothesis that there is no difference between outcomes for those who got the
vaccine versus those who did not. What
is the p-value for this difference?
b.
Create a 95%
confidence interval for the difference between the groups. What is the 90% confidence interval? The 99% interval?
c.
Discuss why this
difference might be observed and how someone might critique the study.
R A Bednarczyk,
R Davis, K Ault, W Orenstein, S B Omer (2012). "Sexual Activity-Related
Outcomes After Human Papillomavirus Vaccination of 11-
to 12-Year-Olds," Pediatrics.
11.
(20 points) Use
the Fed SCF 2010 data (available from Blackboard). This is the Survey of Consumer Finances,
which is not representative (without using the weights, which you need not do
for now) – it intentionally oversamples rich people to find out about their
finances. Concentrate for now on the variable "SAVING" (about the 100th
variable in the list) which is the amount that people have in their savings
accounts.
f.
Test the null
hypothesis that there is no difference between people who are older or younger
than 65. What is the p-value for this
test?
12.
(25 points) Use the ATUS data (available from
Blackboard) on the time that people spend in different activities. Construct a linear regression explaining the
time that people spend on enjoyable activities (t_enjoy
which includes most of the T12 items).
Restrict the data to include only those people spending a non-zero
amount of time on such activities.
g.
What are likely to
be some of the most important determinants of time spent on enjoyable
activities? Which of these are in the
ATUS data? Should the person's wage be
included (do you think income or substitution effect would dominate)? What are some important determinants, that
you could imagine a survey measuring, that are not in the ATUS data? You might find descriptive statistics for the
included variables.
h.
Carefully specify
and estimate a linear regression. What
are the statistically significant coefficients?
Which explanatory variables are most important? Are there surprises? Discuss your results. (You might want to estimate more models or
create additional variables.)
13.
(15 points) You might find it useful to sketch
the distributions.
i.
For a Standard
Normal Distribution, what is the area closer to the mean than 1.45?
j.
For a Standard
Normal Distribution, what is the area to the right of 2?
k.
For a Normal
Distribution with mean
5 and standard deviation 7.6, what is area to the right of 14.1?
l.
For a Normal
Distribution with mean
1 and standard deviation 7.8, what is area in both tails farther
from the mean than 11?
m.
For a Normal
Distribution with mean -5 and standard deviation 1.6, what is area in both
tails farther from the mean than -2.6?
n.
For a Normal
Distribution with mean -1 and standard deviation 9.8, what values leave
probability 0.157 in both tails?
14.
(15 points) In a
medical study (reference below), people were randomly assigned to use either
antibacterial products or regular soap.
In total 592 people used antibacterial soap; 586 used regular soap. It was found that 33.1% of people using
antibacterial products got a cold; 32.3% of people using regular soap got
colds.
d.
Test the null
hypothesis that there is no difference in the rates of sickness for people
using regular or antibacterial soap. (What is the p-value?)
e.
Create a 95%
confidence interval for the difference in sickness rates. What is the 90% confidence interval? The 99% interval?
f.
Every other study
has found similar results. Why do you
think people would pay more for antibacterial soaps?
E.L.Larson, S.X. Lin, C. Gomez-Pichardo,
P. Della-Latta, (2004). "Effect of Antibacterial Home Cleaning
and Handwashing Products on Infectious Disease
Symptoms: A Randomized Double-Blind Trial," Ann Intern Med, 140(5),
321-329.
15.
(15 points) A
study of workers and managers asked both how much management listened to
workers' suggestions (on a scale of 1-7 where "1" indicates that they
paid great attention). Managers averaged
a 2.50 (standard deviation of 0.55); workers answered an average 2.08 (standard
deviation of 0.76) – managers ignore their workers even more often than the
employees realize. There were 137
workers and 14 managers answering.
d.
Test the null
hypothesis that there was no difference between workers and managers: how
likely is it that there is actually no difference in average response? (What is
the p-value?)
e.
Create a 95%
confidence interval for the difference between workers and managers. What is the 90% confidence interval? The 99% interval?
16.
(15 points) A
recent survey by Intel showed that 53% of parents (561 were surveyed) were
uncomfortable talking with their children about math & science. Previous surveys found that 57% of parents
talked with their kids about sex & drugs.
a.
Test the null
hypothesis that parents are as comfortable talking about math & science as
sex & drugs; that the true value of parents uncomfortable with math and science
is not different from 57%. What is the
p-value?
b.
Create a 95%
confidence interval for the true fraction of parents who are uncomfortable with
math & science. What is the 90%
confidence interval? The 99% interval?
17.
(15 points) The
New York Times reported on educational companies that over-sell their products
and gave the example of "Cognitive Tutor" (CT) that helps math
students. The CT students improved by
17.41 (standard deviation of 5.82); the regular students improved by 15.28
(standard deviation of 5.33). There were
153 students in the new program and 102 regular students.
d.
Test the null
hypothesis that there is no difference between regular students and those in
the CT group. What is the p-value for
this difference?
e.
Create a 95%
confidence interval for the difference between regular and CT students. What is the 90% confidence interval? The 99% interval?
18.
(20 points) Use
the ATUS data (available from Blackboard) on the time that people spend in
different activities.
o.
Among households
with kids, what is the average time spent on activities related to kids?
p.
Among households
with kids, how much time to men and women spend on activities related to
kids? Form a hypothesis test for whether
there is a statistically significant difference between the time that men and
women spend with kids. What is the
p-value for the hypothesis of no difference?
What is a 95% confidence interval for the difference in time?
q.
Why do you think
that we would find these results?
Explain (perhaps with some further empirical results from the same data
set).
19.
(25 points) Use
the PUMS data (available from Blackboard) on the residents of NYC. Consider the time (in minutes) spent by
people to travel to work; this variable has name JWMNP.
r.
How many men and
women answered this question? What
variables do you think would be relevant, in trying to explain the variation in
commuting times?
s.
Form a linear
regression with the dependent variable, "JWMNP Travel Time to Work,"
and relevant independent variables.
t.
Which independent
variables have coefficients that are statistically significantly different from
zero?
20.
{{this question
was given in advance for students to prepare with their group} Download (from
Blackboard) and prepare the dataset on the 2004 Survey of Consumer Finances
from the Federal Reserve. Estimate the
probability that each head of household (restrict to only heads of household!)
has at least one credit card. Write up a
report that explains your results (you might compare different specifications,
you might consider different sets of socioeconomic variables, different
interactions, different polynomials, different sets of fixed effects, etc.).
21.
Explain in greater
detail your topic for the final project.
Include details about the dataset which you will use and the regressions
that you will estimate. Cite at least
one previous study which has been done on that topic (published in a refereed
journal).
22.
This question
refers to your final project.
a.
What data set will
you use?
b.
What regression
(or regressions) will you run? Explain
carefully whether the dependent variable is continuous or a dummy, and what
this means for the regression specification.
What independent variables will you include?
c.
What other
variables are important, but are not measured and available in your data
set? How do these affect your analysis?
23.
You want to
examine the impact of higher crude oil prices on American driving habits during
the past oil price spike. A regression
of US gasoline purchases on the price of crude oil as well as oil futures gives
the coefficients below. Critique the
regression and explain whether the necessary basic assumptions hold. Interpret each coefficient; explain its
meaning and significance.
Coefficients(a)
|
Model |
|
Unstandardized Coefficients |
Standardized Coefficients |
t |
Sig. |
|
| B |
Std. Error |
Beta |
||||
|
1 |
(Constant) |
.252 |
.167 |
|
1.507 |
.134 |
| return on
crude futures, 1 month ahead |
.961 |
.099 |
.961 |
9.706 |
.000 |
|
| return on
crude futures, 2 months ahead |
-.172 |
.369 |
-.159 |
-.466 |
.642 |
|
| return on
crude futures, 3 months ahead |
.578 |
.668 |
.509 |
.864 |
.389 |
|
| return on
crude futures, 4 months ahead |
-.397 |
.403 |
-.333 |
-.986 |
.326 |
|
| US gasoline
consumption |
-.178 |
.117 |
-.036 |
-1.515 |
.132 |
|
| Spot Price
Crude Oil Cushing, OK WTI FOB (Dollars per Barrel) |
4.23E-005 |
.000 |
.042 |
1.771 |
.079 |
|
a Dependent Variable:
return on crude spot price
24.
You are in charge
of polling for a political campaign. You
have commissioned a poll of 300 likely voters.
Since voters are divided into three distinct geographical groups, the poll
is subdivided into three groups with 100 people each. The poll results are as follows:
|
|
|
total |
|
A |
B |
C |
|
|
number in favor of candidate |
170 |
|
58 |
57 |
55 |
|
|
number total |
300 |
|
100 |
100 |
100 |
|
|
std. dev. of poll |
0.4956 |
|
0.4936 |
0.4951 |
0.4975 |
Note
that the standard deviation of the sample (not the standard error of the
average) is given.
d.
Calculate a
t-statistic, p-value, and a confidence interval for the main poll (with all of
the people) and for each of the sub-groups.
e.
In simple language
(less than 150 words), explain what the poll means and how much confidence the
campaign can put in the numbers.
f.
Again in simple
language (less than 150 words), answer the opposing candidate's complaint,
"The biased media confidently says that I'll lose even though they admit
that they can't be sure about any of the subgroups! That's neither fair nor accurate!"
25.
Fill in the blanks
in the following table showing SPSS regression output. The model has the dependent variable as time
spent working at main job.
Coefficients(a)
|
Model |
|
Unstandardized Coefficients |
Standardized Coefficients |
t |
Sig. |
|
| B |
Std. Error |
Beta |
||||
|
1 |
(Constant) |
198.987 |
7.556 |
|
26.336 |
.000 |
| Female |
-65.559 |
4.031 |
-.138 |
___?___ |
___?___ |
|
| African-American |
-9.190 |
6.190 |
-.013 |
___?___ |
___?___ |
|
| Hispanic |
17.283 |
6.387 |
.024 |
___?___ |
___?___ |
|
| Asian |
1.157 |
12.137 |
.001 |
___?___ |
___?___ |
|
| Native
American/Alaskan Native |
-28.354 |
14.018 |
-.017 |
-2.023 |
.043 |
|
| Education:
High School Diploma |
___?___ |
6.296 |
.140 |
11.706 |
.000 |
|
| Education:
Some College |
___?___ |
6.308 |
.174 |
14.651 |
.000 |
|
| Education:
4-year College Degree |
110.064 |
___?___ |
.183 |
16.015 |
.000 |
|
| Education:
Advanced degree |
126.543 |
___?___ |
.166 |
15.714 |
.000 |
|
| Age |
-1.907 |
___?___ |
-.142 |
-16.428 |
.000 |
|
a Dependent Variable: Time
Working at main job
26.
Suppose I were to
start a hedge fund, called KevinNeedsMoney Limited
Ventures, and I want to present evidence about how my fund did in the
past. I have data on my fund's returns, Rett, at each time period t, and the returns on
the market, Mktt. The graph below shows the relationship of
these two variables:
a.
I run a univariate OLS regression, 
. Approximately what value would be estimated for the
intercept term, b0? For the slope term, b1?
b.
How would you
describe this fund's performance, in non-technical language – for instance if
you were advising a retail investor without much finance background?
27.
Using the American
Time Use Study (ATUS) we measure the amount of time that each person reported
that they slept. We run a regression to
attempt to determine the important factors, particularly to understand whether
richer people sleep more (is sleep a normal or inferior good) and how sleep is
affected by labor force participation.
The SPSS output is below.
|
Coefficients(a) |
|
|
|
|
|
||
|
Model |
Unstandardized
Coefficients |
Standardized Coefficients |
|
|
|||
|
|
|
B |
Std. Error |
Beta |
t |
Sig. |
|
|
1 |
(Constant) |
-4.0717 |
4.6121 |
|
-0.883 |
0.377 |
|
|
|
female |
23.6886 |
1.1551 |
0.18233 |
20.508 |
0.000 |
|
|
|
African-American |
-8.5701 |
1.7136 |
-0.04369 |
-5.001 |
0.000 |
|
|
|
Hispanic |
10.1015 |
1.7763 |
0.05132 |
5.687 |
0.000 |
|
|
|
Asian |
-1.9768 |
3.3509 |
-0.00510 |
-0.590 |
0.555 |
|
|
|
Native
American/Alaskan Native |
-3.5777 |
3.8695 |
-0.00792 |
-0.925 |
0.355 |
|
|
|
Education:
High School Diploma |
2.5587 |
1.8529 |
0.01768 |
1.381 |
0.167 |
|
|
|
Education:
Some College |
-0.3234 |
1.8760 |
-0.00222 |
-0.172 |
0.863 |
|
|
|
Education:
4-year College Degree |
-1.3564 |
2.0997 |
-0.00821 |
-0.646 |
0.518 |
|
|
|
Education:
Advanced degree |
-3.3303 |
2.4595 |
-0.01590 |
-1.354 |
0.176 |
|
|
|
Weekly
Earnings |
0.000003 |
0.000012 |
-0.00277 |
-0.246 |
0.806 |
|
|
|
Number
of children under 18 |
2.0776 |
0.5317 |
0.03803 |
3.907 |
0.000 |
|
|
|
person
is in the labor force |
-11.6706 |
1.7120 |
-0.08401 |
-6.817 |
0.000 |
|
|
|
has
multiple jobs |
0.4750 |
2.2325 |
0.00185 |
0.213 |
0.832 |
|
|
|
works
part time |
4.2267 |
1.8135 |
0.02244 |
2.331 |
0.020 |
|
|
|
in
school |
-5.4641 |
2.2993 |
-0.02509 |
-2.376 |
0.017 |
|
|
|
Age |
1.1549 |
0.1974 |
0.31468 |
5.850 |
0.000 |
|
|
|
Age-squared |
-0.0123 |
0.0020 |
-0.33073 |
-6.181 |
0.000 |
|
a.
Which variables
are statistically significant at the 5% level?
At the 1% level?
b.
Are there other
variables that you think are important and should be included in the
regression? What are they, and why?
28.
Use the SPSS
dataset, atus_tv from Blackboard, which is a subset
of the American Time Use survey. This
time we want to find out which factors are important in explaining whether
people spend time watching TV. There are
a wide number of possible factors that influence this choice.
a.
What fraction of
the sample spend any time watching TV?
Can you find sub-groups that are significantly different?
b.
Estimate a
regression model that incorporates the important factors that influence TV
viewing. Incorporate at least one
non-linear or interaction term. Show the
SPSS output. Explain which variables are
significant (if any). Give a short
explanation of the important results.
29. Estimate the following regression:: S&P100 returns = b0 + b1(lag S&P100 returns) + b2(lag interest rates) + ε
using the
dataset, financials.sav. Explain which coefficients (if any) are
significant and interpret them.
30.
A study by Mehran
and Tracy examined the relationship between stock option grants and measures of
the company's performance. They
estimated the following specification:
Options = b0+b1(Return on Assets)+b2(Employment)+b3(Assets)+b4(Loss)+u
where the
variable (Loss) is a dummy variable
for whether the firm had negative profits.
They estimated the following coefficients:
|
|
Coefficient |
Standard
Error |
|
Return on
Assets |
-34.4 |
4.7 |
|
Employment |
3.3 |
15.5 |
|
Assets |
343.1 |
221.8 |
|
Loss Dummy |
24.2 |
5.0 |
Which
estimate has the highest t-statistic (in absolute value)? Which has the lowest p-value? Show your calculations. How would you explain the estimate on the
"Loss" dummy variable?
31.
Calculate the
probability in the following areas under the Normal pdf with mean and standard
deviation as given. You might usefully
draw pictures as well as making the calculations. For the calculations you can use either a
computer or a table.
g.
What is the
probability, if the true distribution has mean -15 and standard deviation of
9.7, of seeing a deviation as large (in absolute value) as -1?
h.
What is the
probability, if the true distribution has mean 0.35 and standard deviation of
0.16, of seeing a deviation as large (in absolute value) as 0.51?
i.
What is the
probability, if the true distribution has mean -0.1 and standard deviation of
0.04, of seeing a deviation as large (in absolute value) as -0.16?
32.
Using data from
the NHIS, we find the fraction of children who are female, who are Hispanic,
and who are African-American, for two separate groups: those with and those
without health insurance. Compute tests
of whether the differences in the means are significant; explain what the tests
tell us. (Note that the numbers in
parentheses are the standard deviations.)
|
|
with health
insurance |
without health
insurance |
|
female |
0.4905 (0.49994)
N=7865 |
0.4811 (0.49990)
N=950 |
|
Hispanic |
0.2587 (0.43797)
N=7865 |
0.5411 (0.49857)
N=950 |
|
African
American |
0.1785 (0.38297)
N=7865 |
0.1516 (0.35880)
N=950 |
33.
A paper by Farber
examined the choices of how many hours a taxidriver
would work, depending on a number of variables.
His output is:
"Driver
Effects" are fixed effects for the 21 different drivers.
a.
What is the
estimated elasticity of hours with respect to the wage?
b.
Is there a
significant change in hours on rainy days?
On snowy days?
34.
For the ATUS dataset, use "Analyze \ Descriptive Statistics \
Crosstabs" to create a joint probability table showing the fractions of
males/females about the amount of time spent on the computer vs watching TV (if either or both are above average). Find and interpret the joint probabilities
and marginal probabilities. Do this for
age groups as well.
35.
Calculate the probability in the following areas under the Standard
Normal pdf with mean of zero and standard deviation of one. You might usefully draw pictures as well as
making the calculations. For the
calculations you can use either a computer or a table.
a.
What is the probability, if the true distribution is a Standard
Normal, of seeing a deviation from zero as large (in absolute value) as 1.9?
b.
What is the probability, if the true distribution is a Standard
Normal, of seeing a deviation from zero as large (in absolute value) as -1.5?
c.
What is the probability, if the true distribution is a Standard
Normal, of seeing a deviation as large (in abs0lute value) as 1.2?
36.
Calculate the probability in the following areas under the Normal pdf
with mean and standard deviation as given.
You might usefully draw pictures as well as making the
calculations. For the calculations you
can use either a computer or a table.
a.
What is the probability, if the true distribution has mean -1 and
standard deviation of 1.5, of seeing a deviation as large (in absolute value)
as 2?
b.
What is the probability, if the true distribution has mean 50 and
standard deviation of 30, of seeing a deviation as large (in absolute value) as
95?
c.
What is the probability, if the true distribution has mean 0.5 and
standard deviation of 0.3, of seeing a deviation as large (in absolute value)
as zero?
37.
A paper by Chiappori, Levitt, and Groseclose (2002) looked at the strategies of penalty
kickers and goalies in soccer. Because
of the speed of the play, the kicker and goalie must make their decisions
simultaneously (a Nash equilibrium in mixed strategies). For example, if the goalie moves to the left
when the kick also goes to the left, the kick scores 63.2% of the time; if the
goalie goes left while the kick goes right, then the kick scores 89.5% of the
time. In the sample there were 117
occurrences when both players went to the left and 95 when the goalie went left
while the kick went right. What is the
p-value for a test that the probability of scoring is different? What advice, if any, would you give to kickers,
based on these results? Why or why not?
38.
A paper by Claudia Goldin and Cecelia Rouse (1997) discusses the fraction of
men and women who are hired by major orchestras after auditions. Some orchestras had applicants perform from
behind a screen (so that the gender of the applicant was unknown) while other
orchestras did not use a screen and so were able to see the gender of the
applicant. Their data show that, of 445
women who auditioned from behind a screen, a fraction 0.027 were
"hired". Of the 599 women who
auditioned without a screen, 0.017 were hired.
Assume that these are Bernoulli random variables. Is there a statistically significant
difference between the two samples? What
is the p-value? Explain the possible
significance of this study.
39.
Another paper, by Kristin Butcher and Anne Piehl
(1998), compared the rates of institutionalization (in jail, prison, or mental
hospitals) among immigrants and natives.
In 1990, 7.54% of the institutionalized population (or 20,933 in the
sample) were immigrants. The standard
error of the fraction of institutionalized immigrants is 0.18. What is a 95% confidence interval for the
fraction of the entire population who are immigrants? If you know that 10.63% of the general
population at the time are immigrants, what conclusions can be made? Explain.
40.
Calculate the
probability in the following areas under the Standard Normal pdf with mean of
zero and standard deviation of one. You
might usefully draw pictures as well as making the calculations. For the calculations you can use either a
computer or a table.
a.
What is the
probability, if the true distribution is a Standard Normal, if seeing a value
as large as 1.75?
b.
What is the
probability, if the true distribution is a Standard Normal, if seeing a value
as large as 2?
c.
If you observe a
value of 1.3, what is the probability of observing such an extreme value, if
the true distribution were Standard Normal ?
d.
If you observe a
value of 2.1, what is the probability of observing such an extreme value, if
the true distribution were Standard Normal ?
e.
What are the
bounds within which 80% of the probability mass of the Standard Normal lies?
f.
What are the
bounds within which 90% of the probability mass of the Standard Normal lies?
g.
What are the
bounds within which 95% of the probability mass of the Standard Normal lies?
41.
Consider a
standard normal pdf with mean of zero and standard deviation of one.
a.
Find the area
under the standard normal pdf between -1.75 and 0.
b.
Find the area
under the standard normal pdf between 0 and 1.75.
c.
What is the
probability of finding a value as large (in absolute value) as 1.75 or larger,
if it truly has a standard normal distribution?
d.
What values form a
symmetric 90% confidence interval for the standard normal (where symmetric
means that the two tails have equal probability)? A 95% confidence interval?
42.
Now consider a
normal pdf with mean of 3 and standard deviation of 4.
a.
Find the area
under the normal pdf between 3 and 7.
b.
Find the area
under the normal pdf between 7 and 11.
c.
What is the
probability of finding a value as far away from the mean as 7 if it truly has a
normal distribution?
43.
If a random
variable is distributed normally with mean 2 and standard deviation of 3, what
is the probability of finding a value as far from the mean as 6.5?
44.
If a random
variable is distributed normally with mean -2 and standard deviation of 4, what
is the probability of finding a value as far from the mean as 0?
45.
If a random
variable is distributed normally with mean 2 and standard deviation of 3, what
values form a symmetric 90% confidence interval?
46.
If a random
variable is distributed normally with mean 2 and standard deviation of 2, what
is a symmetric 95% confidence interval?
What is a symmetric 99% confidence interval?
47.
A random variable is
distributed as a standard normal. (You
are encouraged to sketch the PDF in each case.)
a.
What is the probability that we
could observe a value as far or farther than 1.7?
b.
What is the probability that we
could observe a value nearer than 0.7?
c.
What is the probability that we
could observe a value as far or farther than 1.6?
d.
What is the probability that we
could observe a value nearer than 1.2?
e.
What value would leave 15% of
the probability in the left tail?
f.
What value would leave 10% of
the probability in the left tail?
48.
A random variable is
distributed with mean of 8 and standard deviation of 4. (You are encouraged to sketch the PDF in each
case.)
a.
What is the probability that we
could observe a value lower than 6?
b.
What is the probability that we
could observe a value higher than 12?
c.
What is the probability that
we'd observe a value between 6.5 and 7.5?
d.
What is the probability that
we'd observe a value between 5.5 and 6.5?
e.
What is the probability that
the standardized value lies between 0.5 and -0.5?
49.
You know that a random variable
has a normal distribution with standard deviation of 16. After 10 draws, the average is -12.
a.
What is the standard error of
the average estimate?
b.
If the true mean were -11, what
is the probability that we could observe a value between -10.5 and -11.5?
50.
You know that a random variable
has a normal distribution with standard deviation of 25. After 10 draws, the average is -10.
a.
What is the standard error of
the average estimate?
b.
If the true mean were -10, what
is the probability that we could observe a value between -10.5 and -9.5?
51.
You are consulting for a
polling organization. They want to know
how many people they need to sample, when predicting the results of the
gubernatorial election.
a.
If there were 100 people
polled, and the candidates each had 50% of the vote, what is the standard error
of the poll?
b.
If there were 200 people polled?
c.
If there were 400 people
polled?
d.
If one candidate were ahead
with 60% of the vote, what is the standard error of the poll?
e.
They want the poll to be 95%
accurate within plus or minus 3 percentage points. How many people do they need to sample?
52.
Using the ATUS dataset that
we've been using in class, form a comparison of the mean amount of TV time
watched by two groups of people (you can define your own groups, based on any
of race, ethnicity, gender, age, education, income, or other of your choice).
a.
What are the means for each
group? What is the average
difference?
b.
What is the standard deviation
of each mean? What is the standard error
of each mean?
c.
What is a 95% confidence
interval for each mean?
d.
Is the difference statistically
significant?
53.
(15 points) Calculate the probability in the
following areas under the Standard Normal pdf with mean of zero and standard
deviation of one. You might usefully
draw pictures as well as making the calculations. For the calculations you can use either a
computer or a table.
d.
What is the
probability, if the true distribution is a Standard Normal, of seeing a
deviation from zero as large (in absolute value) as 1.9?
e.
What is the
probability, if the true distribution is a Standard Normal, of seeing a deviation
from zero as large (in absolute value) as -1.5?
f.
What is the
probability, if the true distribution is a Standard Normal, of seeing a
deviation as large (in abs0lute value) as 1.2?
54.
(15 points)
Calculate the probability in the following areas under the Normal pdf with mean
and standard deviation as given. You
might usefully draw pictures as well as making the calculations. For the calculations you can use either a
computer or a table.
d.
What is the
probability, if the true distribution has mean -1 and standard deviation of
1.5, of seeing a deviation as large (in absolute value) as 2?
e.
What is the
probability, if the true distribution has mean 50 and standard deviation of 30,
of seeing a deviation as large (in absolute value) as 95?
f.
What is the
probability, if the true distribution has mean 0.5 and standard deviation of
0.3, of seeing a deviation as large (in absolute value) as zero?
55.
(20 points) Below
is some SPSS output from a regression from the ATUS. The data encompass only the group of people
who report that they spent non-zero time in education-related activities such
as going to class or doing homework for class.
The regression examines the degree to which education-time crowds out
TV-watching time. The dependent is time
spent watching TV. The independents are
time spent on all Education-related activities as well as the usual demographic
variables. Fill in the blanks.
Coefficients(a)
|
Model |
|
Unstandardized Coefficients |
Standardized Coefficients |
t |
Sig. |
|
| B |
Std. Error |
Beta |
||||
|
1 |
(Constant) |
160.531 |
14.658 |
|
10.952 |
.000 |
| time spent on
Education-related activities |
-.137 |
.023 |
-.224 |
__?__ |
__?__ |
|
| female |
-26.604 |
7.852 |
-.112 |
__?__ |
__?__ |
|
| African-American |
-4.498 |
__?__ |
-.014 |
-.417 |
.677 |
|
| Hispanic |
__?__ |
12.181 |
-.023 |
-.681 |
.496 |
|
| Asian |
-7.881 |
19.291 |
-.013 |
__?__ |
__?__ |
|
| Native
American/Alaskan Native |
-4.335 |
28.633 |
-.005 |
-.151 |
__?__ |
|
| Education:
High School Diploma |
1.461 |
13.415 |
.004 |
.109 |
__?__ |
|
| Education:
Some College |
3.186 |
__?__ |
.012 |
.311 |
.756 |
|
| Education:
4-year College Degree |
-47.769 |
13.471 |
-.144 |
-3.546 |
__?__ |
|
| Education:
Advanced degree |
__?__ |
18.212 |
-.131 |
-3.379 |
.001 |
|
| Age |
__?__ |
.276 |
.121 |
2.839 |
.005 |
|
| Weekly
earnings [2 implied decimals] |
.000 |
.000 |
-.041 |
-.990 |
.322 |
|
| In the Labor
Force |
-25.210 |
10.794 |
-.107 |
__?__ |
.020 |
|
| Has multiple
jobs |
.918 |
15.299 |
.002 |
__?__ |
.952 |
|
| Works part
time |
3.816 |
10.427 |
.015 |
.366 |
.714 |
|
a Dependent Variable:
watching TV (not religious)
56.
Using the same
SPSS output from the regression above, explain clearly which variables are
statistically significant. Provide an
interpretation for each of the observed signs.
What about the magnitude of the coefficients? What additional variables (that are in the
dataset) should be included? What
results are surprising to you? (Note your answer should be a well-written few
paragraphs, not just terse answers to the above questions.)
57.
Use the CPS
dataset (available from Blackboard) to do a regression. Explain why your dependent variable might be
caused by your independent variable(s).
What additional variables (that are in the dataset) might be
included? Why did you exclude
those? Next examine the regression
coefficients. Which ones are significant? Do the signs match what would be predicted by
theory? Are the magnitudes
reasonable? (Note your answer should be
a well-written few paragraphs, not just terse answers to the above
questions. No SPSS output dumps either!)
58.
A colleague
proposes the following fitted line.
Explain how or if his model could be an OLS regression. There are 100 observations of pairs of 
and for simplicity
assume 
for all
. For the first 99
observations, the fitted value, 
, is equal to the actual value, so 
. But for the 100th
observation the fitted value misses the true value by 2, so 
. If the fitted values
do not come from an OLS regression, how should the fitted model be changed?