The average annual precipitation for a large Midwest City is

The average annual precipitation for a large Midwest City is 30.85 inches with a standard deviation of 3.6 inches. Assume the variable is normally distributed.

a. Find the probability that a randomly selected month will have less than 30 inches.

b. Find the probability that the mean of a random selection of 32 months will have a mean less than 30 inches.

c. Does it seem reasonable that one month could have a rainfall amount less than 30 inches?

d. Does it seem reasonable that the mean of a sample of 32 months could be less than 30 inches?

Solution

a. Find the probability that a randomly selected month will have less than 30 inches.

P(X<30) = P((X-mean)/s <(30-30.85)/3.6)

=P(Z<-0.24)=0.4052 (from standard normal table)

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b. Find the probability that the mean of a random selection of 32 months will have a mean less than 30 inches.

P(xbar<30) = P((xbar-mean)/(s/vn) <(30-30.85)/(3.6/sqrt(32)))

=P(Z<-1.34) =0.0901 (from standard normal table)

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c. Does it seem reasonable that one month could have a rainfall amount less than 30 inches?

Yes, because the probability is greater than 0.05

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d. Does it seem reasonable that the mean of a sample of 32 months could be less than 30 inches?

Yes, because the probability is greater than 0.05