final(fall2002)--solutions

final(fall2002)--solutions - 1 Problem 1(5 pts Multiple...

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Problem 1. (5 pts) Multiple choice. A random sample of size n = 200 was drawn from a population. A histogram of the data is shown in the figure. The sample was most likely drawn from which distribution? A. T with 3 degrees of freedom B. Poisson with λ = 3 C. Skewed normal with μ = 3 D. Chi-squared with 3 degrees of freedom E. Exponential with E [ X ] = 1/3 Problem 2. (5 pts) Multiple choice. Let X 1 , X 2 , X 3 be a random sample from a uniform distribution with unknown mean μ and unknown variance σ 2 . Knowing full well that a valid confidence interval based on the t statistic is based on the assumption that the sample is drawn from a normally distributed population, a probabilist constructs a “100(1– α )% confidence interval” for μ as 3 / 2 , 2 / S t x α ± . Then, ) 3 / 3 / ( 2 , 2 / 2 , 2 / S t X S t X P α α μ + - would be: A. 1 – α B. Less than 1 – α C. Greater than 1 – α Problem 3. (5 pts) A company uses a urine test to screen job applicants for marijuana use. The test correctly gives positive results for 93% of marijuana users and correctly gives negative results for 98% of non-users. Assume that 33% of college students in the U.S. use marijuana, what is the probability that a randomly selected student who tests positive is actually a marijuana user? Let T be the event that the student tests positive. Let U be the event that the student uses marijuana. Given: 93 . ) | ( = U T P , 98 . ) | ( = U T P , and 33 . ) ( = U P . Find: % 8 . 95 ) 67 (. 02 . ) 33 (. 93 . ) 33 (. 93 . ) ( ) | ( ) ( ) | ( ) ( ) | ( ) | ( = + = + = U P U T P U P U T P U P U T P T U P 1 10 5 0 30 20 10 0 X Frequency
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Problem 4. (5 pts) A teller at JFK airport’s foreign exchange window knows that if a customer wishes to convert British pounds ( X ) into American dollars( Y ), the formula for the conversion is Y = 1.64 X – 2. (Each pound is worth $1.64, and there is a $2 service charge.) The average amount presented for exchange is 140 pounds, with standard deviation 48 pounds. Find the standard deviation of the amount of dollars given out. 72 . 78 $ 48 64 . 1 ) ( 64 . 1 ) 2 64 . 1 ( ) ( 2 = = = - = X V X V Y V Problem 5. (15 points total) a. (5 pts) Suppose the number of fish a fisherman catches in an hour is a Poisson random variable with an average rate of 2 fish per hour. If he fishes for one hour tomorrow morning, what is the probability he will catch fewer than 2 fish? Let X be a r.v. for the number of fish caught in an hour. Then, assume X ~ Poisson(2). The probability that he will catch fewer than two fish in one hour would then be: 406 . 0 ! 1 2 ! 0 2 ) 1 ( ) 0 ( ) 2 ( 1 2 0 2 = + = = + = = < - - e e X P X P X P b. (5 pts) What conditions would guarantee that the distribution would be Poisson? — Fish catches independent of each other; and — Average catch rate constant. c. (5 pts) Would you expect those conditions to be met in this case? Explain briefly. No. Morning fish catch rates are likely to depend on weather conditions, so rate would not be constant. Also, catching a fish might disturb the water enough to reduce the probability of catching another in a short time.
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