400Hw07ans - STAT 400 Fall 2011 Homework#7(due Friday...

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STAT 400 Fall 2011 Homework #7 (due Friday, October 21st, by 3:00 p.m.) 1. At Anytown College, the heights of male students are normally distributed with mean 69 inches and standard deviation 4 inches. The heights of female students are also normally distributed with mean 66 inches and standard deviation 3 inches. For Homecoming, a male student and a female student are selected independently at random to be the King and the Queen. What is the probability that the female student selected to be the Queen is taller than the male student selected to be the King? That is, what is the probability P ( King < Queen ) ? Need P ( King < Queen ) = P ( King – Queen < 0 ) = ? E ( King – Queen ) = 69 – 66 = 3. Var ( King – Queen ) = (1) 2 4 2 + (– 1 ) 2 3 2 = 25. SD ( King – Queen ) = 5. ( King – Queen ) is normally distributed. P ( King – Queen < 0 ) = - < 5 3 0 Z P = P ( Z < – 0.60 ) = Φ ( 0.60 ) = 0.2743 .
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2. One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value 20 in. and standard deviation 0.7 in. The length of the second piece is a normal random variable with mean and standard deviation 15 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 32.65 in. and 35.35 in.? Total = First + Second – Overlap. E( Total) = E(First) + E(Second) E(Overlap) = 20 + 15 1 = 34. Var( Total) = Var(First) + Var(Second) + (– 1 ) 2 Var(Overlap) = 0.7 2 + 0.6 2 + 0.2 2 = 0.49 + 0.36 + 0.04 = 0.89. SD( Total) = 89 . 0 = 0.9434. Total has Normal distribution. P( 32.65 < Total < 35.35 ) = - < < - 9434 . 0 34 35 . 35 Z 9434 . 0 34 65 . 32 P = P( – 1.43 < Z < 1.43 ) = 0.9236 – 0.0764 = 0.8472 .
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3. At Sam’s Butcher Shop, ground beef packages vary in weight according to a normal distribution with a mean of 3.1 pounds and a standard deviation of 0.2 pounds, and are sold for $1 per pound. Packages of Bratwurst vary in weight according to a normal distribution with a mean of 2.6 pounds and a standard deviation of 0.3 pounds, and are
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400Hw07ans - STAT 400 Fall 2011 Homework#7(due Friday...

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