Sampling distribution of . Its mean and standard deviation

Just like the sample mean, , the sample proportion is also a random variable. Hence, it possesses a probability distribution, which is called its sampling distribution.

It can be shown by relying on the definition of the mean that the mean value of -that is,the mean of all possible values of is equal to the population proportion p just as the mean of the sampling distribution.

Definition:

The mean of the sample proportion is denoted by and is equal to the population proportion p. Thus,

.

The mean of all possible values is equal to the population proportion p.

Since p, the sample proportion is an unbiased estimator of the population proportion.

Now we are interested in determining the standard deviation of the values. Just as in the case of sample mean, , the standard deviation of depends on whether the sample size is a small proportion of the population or not.

Definition:

The standard deviation of the sample proportion is denoted by and defined as

where

p – is the population proportion,

, and n – is the sample size.

This formula is valid when , where N – is the population size.

If , then is calculated as follows:

,

where is called the finite population correction factor.

5.2.3. Form of the sampling distribution of

Now that we know the mean and standard deviation of , and we want to consider the form of the sampling distribution of . Applying the central limit theorem as it relates to the random variable, we have the following:

Definition:

According to the central limit theorem, the sampling distribution of is approximately normal for a sufficiently large sample size.

The random variable

is approximately distributed as a standard normal.

This approximation is good if .

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