Consider a population with mean = 82 and standard deviation = 12.
(a)
If a random sample of size 64 is selected, what is the probability that the sample mean will lie between 80.8 and 83.2?
(b)
With a random sample of size 100, what is the probability that the sample mean will lie between 80.8 and 83.2?

(a)
We have and . Since n=64 is large, the central limit theorem tells us that the distribution of is approximately normal with

To calculate , we convert to the standardized variable

The z-values corresponding to 80.8 and 83.2 are

and

Consequently,

(b)
We now have n=100, so ,and

Therefore,

Note that the interval (80.8, 83.2) is centered at . The probability that will lie in this interval is larger for n=100 than for n=64.