THE ETHICS OF IMAGES

Who is responsible?

MOVIES

CARTOONS

MERCHANDISE

LITERATURE

MATH 335 Mathematical Statistics I Exam 3 Out-Of-Class [50 pts] -- FALL 2010 -- KSR Due By 4:30pm Monday December 6, 2010 PROBLEMS or QUESTIONS: SEE THE INSTRUCTOR IMMEDIATELY. You may use textbooks, notes, computer packages, calculators, BUT you are not to consult other individuals, except the instructor. ================================================================== 1. (5 pts) A plan for an executive travelers' club has been developed by an airline on the premise that 15% of its current customers would qualify for membership, (that is p = 0.15). 1.a] Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that exactly 2 qualify for membership? Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that at least 2 qualify for membership? 1.b] Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 25 current customers? 1.c] Let X denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the company's premise in favor of the claim that p > 0.15 if x >= 7. What is the probability that the company's premise is rejected when it is actually valid? What is the probability that the company's premise is rejected when in reality p = 0.2 (that is 20% qualify)? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2. (5 pts) The Chrysler corporation is interested in reducing the number of defects in the finish of sheet-molded grille opening panels. (Hsieh and Goodwin, 1986) Suppose, the average number of defects is 0.55 per panel for the current production process. [(a)] Calculate the probability that a panel has at most 2 defect? [(b)] Calculate the probability that a panel has at least 2 defect? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3. (5 pts) On a recent trip I took four flights and encountered eight pilots. Seven of these eight pilots were men, and one was a woman. Let the random variable X represent the number of women in a random sample of 8 commercial pilots. a) Why is it reasonable to model the distribution of X with ... ... a binomial distribution? b) Suppose that half of all commercial pilots are, in fact, women. Determine the probability that I would encounter one or fewer women in a random sample of eight commercial pilots. c) Is the probability in (b) small enough to convince you that fewer than half of all commercial pilots are women? Explain your reasoning. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4. (5 pts) A company (the producer) supplies microprocessors to a manufacturer (the consumer) of electronic equipment. The microprocessors are supplied in batches of 50. The consumer regards a batch as acceptable provided there are not more than 10 defectives in the batch. Rather than test all of the microprocessors in the batch, 10 are selected at random and tested. Suppose that the consumer will accept the batch provided that 2 or less defectives are found in the sample of 10. a) Find the probability that the batch is accepted ... ... when there are 11 defectives in the batch. b) Find the probability that the batch is rejected ... ... when there are 3 defectives in the batch. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5. (5 pts) Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. a) During the season, what is the probability that ... ... Bob makes his second free throw on his fifth shot? b) During the season, what is the probability that ... ... Bob makes his first free throw on his fourth shot? c) During the season, how many free throws do we expect him ... ... to take in order to make his ninth free throw? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6. (5 pts) The mode of a discrete random variable X with p.m.f. p(x) is that value x* for which p(x) is largest (the most probable x value). a) Consider X distributed poisson(lambda) ... and k positive integer >= 1 a.1) Show that P(X = k-1) < P(X = k) if and only if k < lambda. {Note: {Be sure to prove both directions of the "if and only if" in the above! Thus, the poisson distribution is unimodal, ... ... with the mode occuring at floor(lambda). a.2) Show that if lambda is a positive integer then ... ... P(X = lambda) = P(X = lambda - 1). Thus, if lambda is an integer then both lambda and lambda-1 are modes. b) Consider X {# trials to obtain the 1st success}, distributed geometric(theta) ... determine the mode of this distribution. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7. (5 pts) Suppose a certain candy bar lists its weight as 2.13 ounces. Suppose that the actual weights of these candy bars vary according to a normal distribution with mean = 2.2 ounces and stdev = 0.07 ounces. a) What proportion of candy bars weigh less than the advertised weight? b) In a random sample of 5 candy bars, what is the probability that at least one of the candy bars weighs less than the advertised weight? c) If the manufacturer wants to adjust the production process so that only 1 candy bar in 1000 weighs less than the advertised weight, what should the mean of the actual weights be (assuming that the stdev of the weights remains 0.07 ounces)? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8. (5 pts) A gasoline wholesale distributor has bulk storage tanks that hold supplies and are filled every Monday. Of interest to the wholesaler is the proportion of this supply that is sold during the week. Over many weeks of observation, the distributor found that this proportion could be well modeled by a standard beta distribution with alpha = 2 and beta = 2. [a] Find the probability that at least 90% of the supply is sold in a week. [b] Find the probability that less than 50% of the supply is sold in a week. [c] Half of the weeks are below what % of the supply sold? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9. (5 pts) Suppose that Y is a continuous random variable with cumulative distribution function given by F(y) and probability density function f(y). We aften are interested in conditional probabilities of the form P(Y <= y | Y >= c) for a constant c. a) Show that, for y >= c, P(Y <= y | Y >= c) = [F(y) - F(c)]/[1 - F(c)] b) Show that the function in a) has all the properties of a cumulative distribution function c) If the length of life Y for a battery has a Weibull distribution with beta = 2 and alpha = 1/3 (with measurements in years), find the probability that the battery will last less than 4 years, given it is now 2 years old. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 10. (5 pts) A store owner has overstocked a certain item and decides to use the following promotion (hidden from the customers) to decrease the supply. The item has a marked price of $100. For each customer purchasing the item during a particular day, the owner will reduce the advertised price by half. Thus the first customer will pay $100 for the item, the second customer will pay $50, and so on ... Suppose that the number of customers who purchase the item during the particular day has a Poisson distribution with lambda = 2. Determine the expected advertised price of the item at the end of the day.

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