1. Simple random sampling uses a sample of size n from a population of size N to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible?
2. Suppose that we have a sample space S = (E1, E2 E3, E4, E5, E6, E7), where E1, E2, E7 denote the sample points. The following probability assignments apply: P(E1) = .05, P(E2) = .20, P(E3) = .20, P(E4) = .25, P(E5) = .15, P(E6) = .t0, and P(E7) = .05. Let
A= {E1, E4, E6}
B= {E2, E4, E7}
C = (E2, E3, E5, E7}
a. Find P(A), P(B), and P(C).
b. Find A U B and P(A U B).
c. Find A ∩ B and P(A ∩ B).
d. Are events A and C mutually exclusive?
e. Find Bc and P(Bc).
3. Assume that we have two events, A and B, that are mutually exclusive. Assume further that we know P(A) = .30 and P(B) = .40.
a. What is P(A ∩ B)?
b. What is P(A | B)?
c. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer.
d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?
4. The prior probabilities for events A1, and A2 are P(A1) = .40 and P(A2) = .60. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = .20 and P(B | A2) = .05.
a. Are A1, and A2 mutually exclusive? Explain.
b. Compute P(A1, ∩ B) and P(A2 ∩ B).
c. Compute P(B).
d. Apply Bayes' theorem to compute P(A1| B) and P(A 2 | B).
5. Consider the experiment of tossing a coin twice.
a. List the experimental outcomes.
b. Define a random variable that represents the number of heads occurring on the two tosse54
c. Show what value the random variable would assume for each of the experimental outcomes.
d. Is this random variable discrete or continuous?
6. The following table provides a probability distribution for the random variable x.
x f(x)
3 .25
6 .50
9 .25
a. Compute E(x), the expected value of x.
b. Compute σ2, the variance of x.
c. Compute σ, the standard deviation of x.
7. The following table provides a probability distribution for the random variable y.
|
y
|
f(y)
|
|
2
|
.20
|
|
4
|
.30
|
|
7
|
.40
|
|
8
|
.10
|
a. Compute E(y).
b. Compute Var(y) and a.
8. In San Francisco, 30% of workers take public transportation daily (USA Today, Decemb 21, 2005).
a. In a sample of 10 workers, what is the probability that exactly three workers tali public transportation daily?
b. In a sample of 10 workers, what is the probability that at least three workers t public transportation daily?
9. The National Safety Council (NSC) estimates that off-the-job accidents cost U.S. businesses almost $200 billion annually in lost productivity (National Safety Council, March 2006). Based on NSC estimates, companies with 50 employees are expected to average three employee off-the-job accidents per year. Answer the following questions for companies with 50 employees.
a. What is the probability of no off-the-job accidents during a one-year period?
b. What is the probability of at least two off-the-job accidents during a one-year period?
c. What is the expected number of off-the-job accidents during six months?
d. What is the probability of no off-the-job accidents during the next six months?