7.3 The sampling distribution for proportions

The Central Limit Theorem tells us that the point estimate for the sample mean, $\overline{x}$ , comes from a normal distribution of $\overline{x}$ 's. This theoretical distribution is called the sampling distribution of $\overline{x}$ 's. We now investigate the sampling distribution for another important parameter we wish to estimate; p from the binomial probability density function.

If the random variable is discrete, such as for categorical data, then the parameter we wish to estimate is the population proportion. This is, of course, the probability of drawing a success in any one random draw. Unlike the case just discussed for a continuous random variable where we did not know the population distribution of X's, here we actually know the underlying probability density function for these data; it is the binomial. The random variable is X = the number of successes and the parameter we wish to know is p, the probability of drawing asuccess which is of course the proportion of successes in the population. The question at issue is: from what distribution was the sample proportion, $\mathrm{p\text{'}}=\frac{x}{n}$ drawn? The sample size is n and Xis the number of success found in that sample. This is a parallel question that was just answered by the Central Limit Theorem: from what distribution was the sample mean, $\overline{x}$ , drawn? We saw that once we knew that the distribution was the Normal distribution then we were able to create confidence intervals for the population parameter, µ. We will also use this same information to test hypotheses about the population mean later. We wish now to be able to develop confidence intervals for the population parameter "p" from the binomial probability density function.

In order to find the distribution from which sample proportions come we need to develop the sampling distribution of population proportions just as we did for sample means. So again imagine that we randomly sample say 50 people and ask them if they support the new school bond issue. From this we find a sample proportion, p', and graph it on the axis of p's. We do this again and again etc., etc. until we have the theoretical distribution of p's. Some sample proportions will show high favorability toward the bond issue and others will show low favorability because random sampling will reflect the variation of views within the population. What we have done can be seen in [link] . The top panel is the population distributions of probabilities for each possible value of the random variable X. While we do not know what the specific distribution looks like because we do not know p, the population parameter, we do know that it must look something like this. In reality, we do not know either the mean or the standard deviation of this population distribution, the same difficulty we faced when analyzing the X's previously.

[link] places the mean on the distribution of population probabilities as µ = np. Below the distribution of the population values is the sampling distribution of p's. Again the Central Limit Theorem tells us that this distribution is normally distributed just like the case of the sampling distribution for $\overline{x}$ 's. This sampling distribution also has a mean, the means of the p's, and a standard deviation, ${𝛿}_{\mathrm{p\text{'}}}$ .