Reading: Wooldridge Appendix B
Submit any 4 problems out of the following:
1 Suppose that a high school student is preparing to take the SAT exam. Explain why his or her eventual SAT score is properly viewed as a random variable.
2 Let X be a random variable distributed as Normal(5,4). Find the probabilities of the following events (you will need to use Table G.1):
(i) P(X <6).
(ii) P(X > 4).
(iii) P(|X-5|> 1).
3 Much is made of the fact that certain mutual funds outperform the market year after year (that is, the return from holding shares in the mutual fund is higher than the return from holding a portfolio such as the S&P 500). For concreteness, consider a 10-year period and let the population be the 4,170 mutual funds reported in The Wall Street Journal on January 1, 1995. By saying that performance relative to the market is random, we mean that each fund has a 50–50 chance of outperforming the market in any year and that perfor-mance is independent from year to year.
(i) If performance relative to the market is truly random, what is the probability that any particular fund outperforms the market in all 10 years?
(ii) Find the probability that at least one fund out of 4,170 funds outperforms the market in all 10 years. What do you make of your answer?
(iii) How would you find the probability that at least five funds outperform the market in all 10 years? (Hint: use binomial probabilities).
4 Just prior to jury selection for O. J. Simpson’s murder trial in 1995, a poll found that about 20% of the adult population believed Simpson was innocent (after much of the physical evidence in the case had been revealed to the public). Ignore the fact that this 20% is an estimate based on a subsample from the population; for illustration, take it as the true percentage of people who thought Simpson was innocent prior to jury selection. Assume that the 12 jurors were selected randomly and independently from the population (although this turned out not to be true).
(i) Find the probability that the jury had at least one member who believed in Simpson’s innocence prior to jury selection. [Hint: Define the Binomial(12,0.20) random variable X to be the number of jurors believing in Simpson’s innocence.]
(ii) Find the probability that the jury had at least two members who believed in Simpson’s innocence.
Hint: P(X>=2)=1-P(X<=1), and P(X<=1)=P(X=0)+P(X=1).
5 If a basketball player is a 74% free throw shooter, then, on average, how many free throws will he or she make in a game with eight free throw attempts?
6 Suppose that a college student is taking three courses: a two-credit course, a three-credit course, and a four-credit course. The expected grade in the two-credit course is 3.5, while the expected grade in the three- and four-credit courses is 3.0. What is the expected overall grade point average for the semester? (Remember that each course grade is weighted by its share of the total number of units.)
