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Sampling Distribution of the Sample Proportion for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

A population proportion is the proportion or percentage of a population that has a specified characteristic or attribute. The population proportion is represented by the letter p. The sample proportion is the proportion or percentage in the sample that has a specified characteristic or attribute. The sample proportion is represented by either the symbol or . We shall use the symbol p to represent the sample proportion in this text.

EXAMPLE 7.15   The nation's work force at a given time is 133,018,000 and the number of unemployed is 7,355,000. The proportion unemployed is p = = .055 and the jobless rate is 5.5%. A sample of size 65,000 is selected from the nation's work force and 3900 are unemployed in the sample. The sample proportion unemployed is = .06 and the sample jobless rate is 6%.

The population proportion is a parameter measured on the complete population and is constant over some time interval. The sample proportion p is a statistic measured on a sample and is considered to be a random variable whose value is dependent on the sample chosen. The set of all possible values of a sample proportion along with the probabilities corresponding to those values is called the sampling distribution of the sample proportion.

EXAMPLE 7.16   According to recent data, the nation's five most popular theme parks are shown in Table 7.8. The table gives the name of the theme park and indicates whether or not the attendance exceeds 10 million per year.

For this population of size N = 5, the proportion of theme parks with attendance exceeding 10 million is p = = .60 or 60%. There are 5 samples of size 4 possible when selected from this population. These samples, the theme parks exceeding 10 million (yes or no), the sample proportion, and the probability associated with the sample proportion, are given in Table 7.9.

For each sample, the sampling error, | p – |, is either .10 or .15. From Table 7.9, it is seen that the probability associated with sample proportion .75 is .2 + .2 = .4 and the probability associated with sample proportion .50 is .2 + .2 + .2 = .6. The sampling distribution for is given in Table 7.10.

For larger populations and samples, the sampling distribution of the sample proportion is more difficult to construct, but the technique is the same.

Mean and Standard Deviation of the Sample Proportion

Since the sample proportion has a distribution, it is a random variable and has a mean and a standard deviation. The mean of the sample proportion is represented by the symbol and the standard deviation of the sample proportion is represented by . The standard deviation of the sample proportion is called the standard error of the proportion. Example 7.17 illustrates how to find the mean of the sample proportion and the standard error of the proportion.

EXAMPLE 7.17   For the sampling distribution of the sample proportion shown in Table 7.10, the mean is

The variance of the sample proportion is

The standard error of the proportion, , is equal to = 0.122.

The relationship between the mean of the sample proportion and the population proportion is expressed by

The standard error of the sample proportion is related to the population proportion, the population size, and the sample size by

The term is called the finite population correction factor. If the sample size n is less than 5% of the population size, i.e., n < .05N, the finite population correction factor is very near one and is omitted in formula (7.8).

If n < .05N, the standard error of the proportion is given by formula (7.9), where q = 1 – p.

EXAMPLE 7.18   In Example 7.16, dealing with the five most popular theme parks, it was shown that p = 0.6 and in Example 7.17, it was shown that = 0.6 illustrating formula (7.7). In Example 7.17 it was also shown that = = 0.122. To illustrate formula (7.8), note that

EXAMPLE 7.19   Suppose 80% of all companies use e-mail. In a survey of 100 companies, the standard error of the sample proportion using e-mail is = . The finite population correction factor is not needed since there is a very large number of companies, and it is reasonable to assume that n < .05N.

Practice problems for these concepts can be found at:

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