20.7 Summary: Finding a CI for $p$

The procedure for computing a confidence interval (CI) for a proportion is:

• Compute the sample proportion, $\hat{p}$, and identify the sample size $n$.
• Compute the standard error, which quantifies how much the value of $\hat{p}$ varies from one sample to the next:

$$\text{s.e.}\left(\hat{p}\right)=\sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}.$$

• Find the multiplier: this is $2$ for an approximate 95% CI using the 68–95–99.7 rule. (Note: (Multiplier$\times$standard error) is called the margin of error.)
• Compute:

$$\hat{p}\pm (\text{Multiplier}\times \text{standard error}).$$

• Check the statistical validity conditions are satisfied.

You must use proportions in this formula, not percentages (that is, values like 0.23 and not 23%).

Example 20.7 (NHANES data) For the NHANES data, first seen in Sect. 12.10, the unknown parameter is $p$, the population proportion of Americans that currently smoke.

In the study, 1466 out of the 3211 respondents who reported their smoking status said they currently smoked: $\hat{p}=1466÷3211=0.4566$.

What is the population proportion $p$ that currently smoke? We don’t know, and the estimate of $p$ from every sample is likely to be different. The standard error is $\text{s.e.}\left(\hat{p}\right)=0.00879$, so the approximate 95% CI for $p$ is $0.4566\pm 0.01758$, or from 0.439 to 0.474. (Check the calculations!)

For the conclusions to be statistically valid, the number of smokers must exceed 5, and the number of non-smokers must exceed 5. Both are true. The CI appears to be statistically valid.

We write:

Based on the sample, we are approximately 95% confident that the interval from from 0.429 to 0.474 straddles the population proportion of smokers in the USA.