7.3: Large Sample Estimation of a Population Proportion

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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 $$[0,1]$$.

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