The amount of time the university professors devote to their jobs per week is normally distributed with a mean of 52 hours and a standard deviation of 6 hours. Find the probability that if three professors are randomly selected all three work for more than 60 hours per week.  

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We are given that the average number of hours worked is 52 hours with a standard deviation of 6 hours. We are asked to find the probability that three randomly selected professors each work more than 60 hours.

(1) First we find the probability that a randomly chosen person works...

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We are given that the average number of hours worked is 52 hours with a standard deviation of 6 hours. We are asked to find the probability that three randomly selected professors each work more than 60 hours.

(1) First we find the probability that a randomly chosen person works more than 60 hours.

Assuming that the hours are normally distributed, we can convert to a standard normal score:

`z=(X-mu)/sigma=(60-52)/6=1.bar(3)`

Now we can use the standard normal table (or some utility) to find the probability that a randomly chosen person exceeds 60 hours:

`P(X>60)=P(z>1.33)~~.0918` (Using the rounded value of 1.33; using the exact value yields .0912)

(2) Now we assume that the hours each professor works is independent, so the probability of three such professors working more than 60 hours is `.0918^3~~.00077`

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