STAT 400 1 Discussion 9 Spring 2015 One piece of PVC pipe is to be inserted inside another piece The length of the first piece is normally distributed with mean value 25 in and standard deviation 0 9 in The length of the second piece is a normal random variable with mean and standard deviation 20 in and 0 6 in respectively The amount of overlap is normally distributed with mean value 1 in and standard deviation 0 2 in Assuming that the lengths and amount of overlap are independent of one another what is the probability that the total length after insertion is between 43 45 in and 45 65 in 2 Consider a 6 pack of soda Suppose that the amount of soda in each can follows a normal distribution with mean 12 06 oz and standard deviation 0 15 oz Assume that all cans are filled independently of each other Find the probability of the following a a can is underfilled i e there is less than 12 oz of soda in a can b all 6 cans are underfilled c at least one of the 6 cans is underfilled d exactly 2 of the 6 cans are underfilled e the average amount of soda in these 6 cans is less than 12 oz 3 5 6 4 Approximate P 39 75 X 41 25 where X is the mean of a random sample of size 32 from a distribution with mean 40 and variance 2 8 4 5 6 8 Let X equal the weight in grams of a miniature candy bar Assume that E X 24 43 and 2 Var X 2 20 Find a E X b Var X c P 24 17 X 24 82 approximately 5 6 4 1 Let X 1 X 2 X n be a random sample from N 2 6 unknown known is the MLE for Show that X 6 4 2 Let X 1 X 2 X n be a random sample from N 2 Show that known 2 1 n n i 1 X i 2 unknown is the MLE for 2 1 One piece of PVC pipe is to be inserted inside another piece The length of the first piece is normally distributed with mean value 25 in and standard deviation 0 9 in The length of the second piece is a normal random variable with mean and standard deviation 20 in and 0 6 in respectively The amount of overlap is normally distributed with mean value 1 in and standard deviation 0 2 in Assuming that the lengths and amount of overlap are independent of one another what is the probability that the total length after insertion is between 43 45 in and 45 65 in Total First Second Overlap E Total E First E Second E Overlap 25 20 1 44 Var Total Var First Var Second 1 2 Var Overlap 0 9 2 0 6 2 0 2 2 0 81 0 36 0 04 1 21 SD Total 1 21 1 1 Total has Normal distribution 45 65 44 43 45 44 P 43 45 Total 45 65 P Z P 0 50 Z 1 50 1 1 1 1 0 9332 0 3085 0 6247 2 Consider a 6 pack of soda Suppose that the amount of soda in each can follows a normal distribution with mean 12 06 oz and standard deviation 0 15 oz Assume that all cans are filled independently of each other Find the probability of the following 12 06 a 0 15 a can is underfilled i e there is less than 12 oz of soda in a can 12 12 06 P X 12 P Z P Z 0 40 0 40 0 3446 0 15 b all 6 cans are underfilled 0 3446 6 0 001675 c OR 6 0 6 C 6 0 3446 0 6554 0 001675 at least one of the 6 cans is underfilled 1 P none 1 0 6554 6 0 920743 d exactly 2 of the 6 cans are underfilled 2 4 6 C 2 0 3446 0 6554 0 32866 e the average amount of soda in these 6 cans is less than 12 oz Need P X 12 n 6 Normal distribution Case 2 12 12 06 P X 12 P Z P Z 0 98 0 98 0 1635 0 15 6 3 5 6 4 4 5 6 8
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