# 7.3: Large Sample Estimation of a Population Proportion

- Page ID
- 565

Learning Objectives

- To understand how to apply the formula for a confidence interval for a population proportion.

Since from Section 6.3, we know the mean, standard deviation, and sampling distribution of the sample proportion \(\hat{p}\), the ideas of the previous two sections can be applied to produce a confidence interval for a population proportion. Here is the formula.

Large Sample \(100(1−\alpha)\%\) Confidence Interval for a Population Proportion

\[\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

A sample is large if the interval \([p-3\sigma_{\hat{p}},p+3\sigma _{\hat{p}}]\) lies wholly within the interval \([0,1]\).

In actual practice the value of \(p\) is not known, hence neither is \(\sigma_{\hat{p}}\). In that case we substitute the known quantity \(\hat{p}\) for \(p\) in making the check; this means checking that the interval

\[\left [ \hat{p}-3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\: \hat{p}+3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right ]\]

lies wholly within the interval \([0,1]\).

Example \(\PageIndex{1}\)

To estimate the proportion of students at a large college who are female, a random sample of \(120\) students is selected. There are \(69\) female students in the sample. Construct a \(90\%\) confidence interval for the proportion of all students at the college who are female.

**Solution**:

The proportion of students in the sample who are female is

\[ \hat{p} =69/120=0.575 \nonumber\]

Confidence level \(90\%\) means that \(\alpha =1-0.90=0.10\) so \(\alpha /2=0.05\). From the last line of Figure 7.1.6 we obtain \(z_{0.05}=1.645\).

Thus

\[\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.575\pm 1.645\sqrt{\frac{(0.575)(0.425)}{120}}=0.575\pm 0.074 \nonumber\]

One may be \(90\%\) confident that the true proportion of all students at the college who are female is contained in the interval \((0.575-0.074,0.575+0.074)=(0.501,0.649)\).

## Summary

- We have a single formula for a confidence interval for a population proportion, which is valid when the sample is large.
- The condition that a sample be large is not that its size \(n\) be at least \(30\), but that the density function fit inside the interval \([0,1]\).