SOLUTION: For problem 5, use the appropriate formula: z = x−μ or z = x−μx . σxσn Suppose the heights of 40-year old women are approximately distributed, w

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Question 1090496: For problem 5, use the appropriate formula: z = x−μ or z = x−μx . σxσn
Suppose the heights of 40-year old women are approximately distributed, with mean 62 inches and a standard deviation of 3 inches.
(a) What is the probability that a 40-year old woman selected at random is between 61 and 63 inches tall?

(b) If a random sample of twenty-five 40-year old women is selected, what is the probability that the mean x height is between 61 and 63 inches?
(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
z=(x-mean)/sd
for a (61-62)/3 is -1/3 and (63-62)/3=+(1/3)
probability z is between -1/3 and 1/3=0.2611
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for b, the standard deviation is sigma/sqrt(n)=3/sqrt(25)=0.6
now the z is between -1/.6 and 1/.6 or -5/3 and + 5/3 or 0.9044
It is much higher. The likelihood of a single woman being outside the range is relatively low, but the likelihood of the mean of 25 of them being outside the range is far less. Any outlier at one end is likely to be counterbalanced by the same on the other end.