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7-5 SAMPLING DISTRIBUTIONS OF MEANS 239

population is required to be 300 milliliters. An engineer takes a random sample of 25 cans and
computes the sample average fill volume to be x 298 milliliters. The engineer will probably
decide that the population mean is 300 milliliters, even though the sample mean was
298 milliliters because he or she knows that the sample mean is a reasonable estimate of and
that a sample mean of 298 milliliters is very likely to occur, even if the true population mean is
300 milliliters. In fact, if the true mean is 300 milliliters, tests of 25 cans made repeatedly,
perhaps every five minutes, would produce values of that vary both above and below
x
300 milliliters.
The sample mean is a statistic; that is, it is a random variable that depends on the results
obtained in each particular sample. Since a statistic is a random variable, it has a probability
distribution.

Definition
The probability distribution of a statistic is called a sampling distribution.

For example, the probability distribution of X is called the sampling distribution of the
mean.
The sampling distribution of a statistic depends on the distribution of the population, the
size of the sample, and the method of sample selection. The next section presents perhaps the
most important sampling distribution. Other sampling distributions and their applications will
be illustrated extensively in the following two chapters.

7-5 SAMPLING DISTRIBUTIONS OF MEANS

Consider determining the sampling distribution of the sample mean . Suppose that a random
X
2
sample of size n is taken from a normal population with mean and variance . Now each
observation in this sample, say, X , X , p , X , is a normally and independently distributed
n
2
1
2
random variable with mean and variance . Then by the reproductive property of the
normal distribution, Equation 5-41 in Chapter 5, we conclude that the sample mean
X p X
X 1 2 n
X
n
has a normal distribution with mean
p
n
X
and variance

2
2
p 2 2
2

X n 2 n
If we are sampling from a population that has an unknown probability distribution, the
sampling distribution of the sample mean will still be approximately normal with mean and