Problem 1) Bob and John are traveling from city A to city D. Bob decides to take the upper route (through B) whereas John takes the lower route (though C) as shown in the following figure:

The travel times (in hours) between the cities indicated are normally distributed as follows:

T1 ∼ N(6,4),T2 ∼ N(4,1),T3 ∼ N(5,9),T4 ∼ N(4,1)

The travel times for T1 and T2 are independent, but T3 and T4 are dependent with covariance 2.4.

a) Which route (upper or lower) should be taken if one wishes to minimize the expected travel time from A to D? Justify.

b) What is the probability that Bob will not arrive in city D within ten hours?

c) What is the probability that John will not arrive in city D within ten hours?

d) What is the probability that Bob will arrive earlier than John?

Problem 2) A survey was recently done and it was found that of the 120 households who were randomly contacted, 20 of them did not own   a television set. Based on this information our estimate of the proportion of household's without a television is   0.1667.

The New York Post uses the survey data to make the claim that   less than 20% of American households don't own a television set. The New York Times uses the same survey data to make the claim that less than 25% of American households don't own a television set. Which claim is supported more by the observed   evidence?

Justify. [Hint we want you to compare Ho : p = .2 Ha : p = .2 and Ho: p = .25 Ha: p < .25]

Problem 3) Let θ be some number between 0 and 1 and   define the following discrete probability distribution for the random variable X:

a) Show that this is a legal probability distribution.

b) We want to estimate θ based on one observation.  Suppose we define our estimator to be

Show that this is an unbiased estimator of θ.

             4       x = 2

T(x)

             0       x ≠ 2

Problem 4) A 95% confidence interval for a population proportion calculated using data from a random sample of size n = 500 is (0.13, 0.73). Which of the following is the margin of error of this interval?

Problem 5) When two Labrador retrievers breed, theory says there should be 50% black, 40% yellow and 10% chocolate offspring. Two Labrador retrievers are bred and a litter consisting of 3 black dogs, 5 yellow dogs, and 2 chocolate dogs is produced. For a goodness of fit test, the χ² statistic would be: