4. Distribution of a Sample Proportion

In this part you will learn about the distribution of sample proportions. In a sense, this is an exact repeat of the sampling distribution of the sample means. Whereas the mean of a population is obtained by averaging the value of interest, a proportion is simply the percent of a population that does or does not have a certain characteristic. The value of a proportion must, therefore, fall between 0 and 1, inclusive: 0 ≤ p ≤ 1.

Our goal, once again, is to use our sample data to predict something about the population. In this case, we’ll use the proportion of our sample that satisfies some specific condition to predict the overall proportion of the population that satisfies the same condition. We denote the population proportion by p, and the sample proportion by .

Calculating a Sample Proportion

To calculate the value of  from a sample of size n, simply count the number of people, x, in the population that satisfy the required condition and divide by the size of the sample, n. In symbols:

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The Sampling Distribution of the Sample Proportion

Just as with the sample mean, the larger our sample size, the more closely will be to the true population proportion p. But since there is randomness to every sample obtained, the value of will vary from sample to sample. Thus, the value of  is a random variable, and must have a mean and standard deviation. It turns out that the distribution of the sample proportion will be approximately normal (as long as the sample size is large enough) with mean (or expected value) of p (the true population proportion) and standard deviation:

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In symbols, the distribution of the sample proportion  is approximately normal with distribution

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It turns out this distribution of the sample proportion holds only when the sample size satisfies an important size requirement, namely that the sample size n be less than or equal to 5% of the population size, N. So n ≤ 0.05 ⋅ N. Although important, in this class we will not focus on this result.

Example (from Fundamentals of Statistics, by Sullivan)

In a survey, 500 parents were asked about the importance of sports for both boys and girls. Of the parents interviewed, 60% agreed that the genders are equal and should have opportunities to participate in sports. Describe the sampling distribution of the sample proportion  of parents who agree that the genders are equal and should have equal opportunities.

We will assume that the sample of 500 represents a random sample of the parents of all boys and girls in the United States. The true proportion in the population is equal to some unknown value . The sampling distribution of  can be approximated by a normal distribution with distribution

section7-10Even though we do not know the exact value of p, we can use  as a good estimator of of p, to approximate the distribution of the sample proportion of

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The distribution of the sample proportion  is approximately normal with distribution N(0.6, 0.022).