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 number
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 situational 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 distribution, treat the data as coming from a continuous distribution even
though they appear discrete.)
Problem 3. (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 λ). 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?