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ST 558 Probability and Statistics Exam # 2 Possible points: 100 [SHOW ALL OF YOUR WORK FOR PARTIAL CREDIT] 1. The width of a slot of a duralumin

ST 558 Probability and Statistics Exam # 2 Possible points: 100 [SHOW ALL OF YOUR WORK FOR PARTIAL CREDIT] 1. The width of a slot of a duralumin forging is ( in inches) normally distributed with μ = 0 . 9000 and σ = 0 . 0030 . The speciﬁcation limits were given as 0 . 9000 ± 0 . 0050 . (a) What percentage of forgings will be defective? (b) What is the maximum allowable value of σ that will permit no more than 1 in 100 defectives when the widths are normally distributed with μ = 0 . 9000 and σ ? 2. One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that number 6 will appear between 150 and 200 times. If number 6 appears exactly 200 times, ﬁnd the probability that number 5 will appear less than 150 times. 3. (a) Suppose that X is a normal random variable with mean 5. If P ( X > 9) = 0 . 2 , ﬁnd the variance of X. (b) The average life of a bread-making machine is 7 years, with a stan- dard deviation of 1 year. Assuming that the lives of these machines follow approximately a normal distribution , ﬁnd (i) the probability that the mean life of a random sample of 9 such machines falls between 6.4 and 7.2 years (ii) For what value to the right of which 15% of the means computed from random samples of size 9 would fall. 4. In an experiment with a new tranquilizer, the pulse rates (per minute) of 12 patients were determined before they were given the tranquilizer and again 5 minutes later, and their pulse rates were found to be re- duced on the average by 7.2 beats with a standard deviation of 1.8. At the level of signiﬁcance 0.05, do we have signiﬁcance evidence that the mean pulse reduction with this tranquilizer is less than 9.0 beats? Also, ﬁnd 95% conﬁdence interval for the average pulse rate. 1
5. A medical research team conducted a study to test the eﬀect of a choles- terol reducing medication. At the end of the study, the researchers found that of the 4700 subjects who took the medication, 301 died of heart disease. Of the 4300 subjects who took a placebo, 357 died of heart disease. At α = 0 . 01 , can you conclude that the death rate is lower for those who took the medication than for those who the placebo? Construct a 95% conﬁdence interval for the two population proportions. 6. A manufacturer claims that the calling range (in miles) of its 900-MHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 phones from the manufacturer had a mean 1275 with a standard deviation of 45. The mean for 16 similar phones from its competitor was 1250 with a standard deviation 30. At α = 0 . 05 , is there enough evidence to support the manufacturer’s claim? Assume the population variances are equal. Find a 95% conﬁdence interval for the diﬀerence of two population means. 7. The random variable X has probability density function f ( X ) = ± aX + bX 2 for 0 < X < 1 0 elsewhere If E(X) =0.6, then (a) Find constants a and b (b) Find P ( X 0 . 5) . (c) Find V ar ( X ) . 8. The joint density of X and Y is given by f ( X, Y ) = ± C ( Y - X ) e - Y for - Y < X < Y, 0 < Y < 0 elsewhere (a) Find C. (b) Find marginal density functions for X and Y. (c) Find E(X) and E(Y). Hint: Γ( α ) = R 0 e - u u α - 1 du and Γ( α ) = ( α - 1)! 2
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