Directions: This is an open-book and open-note take-home exam. Do not use any web sites for in-
formation nor communicate with anyone else about the content of this exam. The ethics statement,
provided in the syllabus, is in effect. Please submit exam by the due date to Blackboard. Late exams
will not be accepted for any reason.
Problem 1. (9 points) World Cup Recap. Recall that in the Group Play portion of the World Cup, there
are eight groups (A-H) of four teams each. Each team plays each of the other teams in its group exactly
once in a round-robin format; as a result, there are six games per group. The total number of goals that
were scored in each game is provided in the table below.
Group A B C D E F G H
Game 1 4 6 3 4 3 3 4 3
Game 2 1 4 3 3 3 0 3 2
Game 3 0 5 3 3 7 1 4 1
Game 4 4 2 0 1 3 1 4 6
Game 5 5 2 5 1 3 5 3 1
Game 6 4 3 3 0 0 4 1 2
(a) Provide three estimates of the population variance of the number of goals scored per game: an
overall estimate, a within estimate, and a between estimate.
(b) Is there any evidence that the group is independent of the game number as it pertains to the num-
ber of total goals scored?
(c) Jack and Jill each perform. a hypothesis test of the null that the mean number of goals scored is
the same for each group. When Jack performs his test, he ignores the in- formation about the game
number. Jill, however, thinks the game number may be relevant; she tells Jack that earlier games
might have more scoring because the players are rested, or, alternatively, maybe less scoring because
the other teams have had an opportunity to learn from observing how their opponents played in the
earlier games. Therefore, Jill wants to use the information about the game number in her test. Do they
come to different conclusions? Fully justify your answer.
Problem 2. (6 points) Robin Williams, the comedic actor, rocketed to fame with his portrayal of a
zany alien on the television show Mork and Mindy. The file Sitcom durations.RData provides a list of
the 760 situational comedies that have aired on US television since 1949. You will see that some shows
aired for many years but most lasted only one season. Do the data suggest that the duration of a situ-
ational comedy (i.e., the number of years it airs) is better described by an exponential distribution or
by a Poisson distribution? Fully justify your answer. (Note: when considering the exponential distri-
bution, treat the data as coming from a continuous distribution even though they appear discrete.)
Problem 3. (8 points). Two Small Samples. Consider two different populations from which you have
collected the following sample data:
Sample 1:f4.1, 1.3, 2.9, 7.3, 7.3, 5.9, 2.6, 1.2, 8.3, 5.9g
Sample 2:f9.4, 5.7, 6.5, 9.9, 6.2, 2.0, 0.7, 5.9, 1.8, 6.3g
Sample 1 was drawn from Population 1; Sample 2 was drawn from Population 2.
(a) Consider two confidence intervals for the difference in the population means. Which is wider: a
95% confidence interval assuming the samples are matched, or a 90% confidence interval assuming
the samples are unmatched?
(b) Using = 0:05, perform. a hypothesis test of the null that the variance of Population 1 is equal to
the variance of Population 2. (c) Consider a hypothesis test of the null that the mean of Population
1 is less than 5. Which yields a smaller probability of Type II error: a test in which is known and
= 0:01, or a test in which is unknown and = 0:05?
(d) Consider a hypothesis test of the difference between the proportion of Population 1 that is less than
3 and the proportion of Population 2 that is less than 3. Which null hypothesis is more easily rejected:
one in which the null is that the difference between those proportions is greater than .10, or one in
which the null is that the difference between those proportions is less than .10?
Problem 4. (10 points) Single-Parameter Estimation. X is distributed according to the pdf
f(x) = 3 exp x 0 x 1 0 < < 10 (1)
(a) What is the maximum likelihood estimate MLE given the sample
f0.6, 0.6, 0.1, 0.8, 0.5, 0.4, 0.5, 0.9, 0.2, 0.4g?
(b) Calculate an estimator for the variance of MLE.
(c) What is the method of moments estimator MOM?
(d) Predict P(X < 3) using MLE estimator?
(e) Predict P(2 Problem 5. (7 points) Job Hunt. Jane and John are both looking for jobs. The number of job offers per
month that each receives is distributed according to a Poisson. Jane has been job searching for a while
and knows that she receives an average of two job offers per month. John has just started looking for a
job and does not know the average number of jobs per month he will receive (call it ). Jane and John
are in different fields and therefore you can assume that the number of job offers that Jane receives is
independent of the number of job offers that John receives.
(a) Create a joint probability table of the number of job offers per month that Jane receives and the
number of job offers that John receives. For each, you should consider the possibilities of 0, 1, and 2 or
more job offers.
(b) Calculate the covariance between the number of offers per month received by Jane and the number
of offers per month received by John.
(c) Provide the cumulative distribution function of the joint probability found in (a).
(d) Consider 100 people like Jane who are looking for work and receive an average of two job offers
per month. On August 30th, those 100 people will be asked how many job offers they received during
the month of August. What is the probability that the average number of job offers received (averaged
over these 100 people) is less than one?
Problem6. Extra-Credit. (6points) Load data Revenue. Run a linear regression model predicting vari-
ation in revenue as a function of (lag.quarterly.revenue, price.index, income.level, market.potential).
(a) Report estimated coefficients on price.index and income.level.
(b) Report 95% confidence intervals on price.index and income.level.
(c) Predict the effect of revenue if price.index increase by one standard deviation and income.level de-
creases by one standard deviation.