Math 251, 28 October 2003, Exam 2
Name: .
Instructions: Complete each of the following seven questions, and please explain and justify all appropriate details in your solutions in order to obtain maximal credit for your answers.
1. At Kenwood College of Engineering, 45% of incoming freshmen students are female and 55% are male. Recent records indicate that 80% of the entering female students will graduate with a BSE degree, while 70% of the male students will obtain a BSE degree. If an incoming engineering student is selected at random, find
(a) (1 pt) P(student will graduate, given student is female)
(b) (2 pts) P(student will graduate, and student is female)
(c) (1 pt) P(student will graduate, given student is male)
(d) (2 pts) P(student will graduate, and student is male)
(e) (2 pts) P(student will graduate)
(f) (2 pts) P(student will graduate, or student is female)
2. (a) (2 pts) President Geraty has recently received permission to excavate the site of an ancient temple. In how many ways can he choose 8 of the 29 students in his History of Antiquities course to join him on the dig?
(b) (2 pts) Of the 29 graduate students, 19 are female and 10 are male. In how many ways can President Geraty select a group of 8 that consists only of females?
(c) (2 pts) What is the probability that President Geraty would randomly select a group of 8 consisting of only females?
3. (a) (1 pt) Explain what mutually exclusive events are.
(b) (1 pt) Explain what independent events are.
(c) (2 pts) Give an example of two events that independent but are not mutually exclusive. Justify your answer.
4. (a) (1 pt) Fill in the missing probability for the following discrete random variable:
x / 3 / 6 / 9 / 10 / 12P(x) / .10 / .15 / .25 / ? / .11
? = .
(b) The number of cars per household in a small town is given by
Cars 0123
Households 20 280 75 25
(i) (3 pts) Make a probability distribution for x where x represents the number of cars per household in this small town.
Cars (x)P(x)
(ii) (5 pts) Find the mean and standard deviation for the random variable in (i)
(iii) (1 pts) What is the average number of cars per household in that small town? Explain what you mean by average.
5. Suppose a baseball player has a batting average of 0.280 (the probability of getting a hit at an at-bat). Suppose the player had 12 at-bats in a weekend series.
(a) (2 pts) What is the probability that the player had no hits?
(b) (2 pts) What is the probability that the player had exactly one hit?
(c) (2 pts) What is the probability that the player had 2 or more hits?
(d) (4 pts) Find the mean and standard deviation for the number of hits the player will get in 400 at-bats.
6. Suppose the distribution of weights of adult male American Landrace pigs is normally distributed with a mean of 480 lbs and standard deviation of 55 lbs.
(a) (2 pts) What weight is at the 90th percentile?
(b) (2 pts) What proportion of adult male American Landrace pigs weigh between 500 lbs and 600 lbs?
(c) (2 pts) What proportion of adult male American Landrace pigs weigh less than 600 lbs?
(d) (2 pts) What proportion of adult male American Landrace pigs weigh more than 600 lbs?
(e) (2 pts) Find an interval whose center is the mean, and which contains 99% of the adult male
American Landrace pigs weights. Hint: first find a z-interval so that 99% of normal curve
lies between –z and z.
7. Alaska Airlines has found that 92% of people with tickets will show up for their Friday afternoon flight from Seattle to Ontario. Suppose that there are 185 passengers holding tickets for this flight, and the jet can carry 178 passengers, and that the decisions of passengers to show up are independent of one another.
(a) (2 pts) Verify that the normal approximation of the binomial distribution can be used for this problem.
(b) (3 pts) What is the probability that 178 or fewer passengers will show up for the flight (i.e., all passengers who show up will receive a seat)?