Applied Statistics and Probability for Engineers

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 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 

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.

x

x 298

For example, the probability distribution of 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

sample of size nis taken from a normal population with mean and variance ^2. Now each

observation in this sample, say, X 1 , X 2 ,, Xn, is a normally and independently distributed

random variable with mean and variance ^2. Then by the reproductive property of the

normal distribution, Equation 5-41 in Chapter 5, we conclude that the sample mean

has a normal distribution with mean

and variance

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

X^2 

^2 ^2 p^2

n^2



^2

n

X

p

n 

X

X 1 X 2 pXn

n

p

X

X

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

Definition

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