20.2 Sampling intervals: Known proportion

The possible values of the sample proportions $\hat{p}$ can be described by an approximate normal distribution, as just discussed. This enables the 68–95–99.7 rule to be applied; for example, about 68% of the time with sets of 25 rolls, the sample proportion of even rolls will be between $0.5$ give-or-take one standard deviation (that is, give-or-take 0.1). So, about 68% of the time, the proportion of even rolls in a set of 25 rolls will be between:

• $0.5-0.1=0.4$ and
• $0.5+0.1=0.6$.

Similarly, about 95% of the time, the proportion of even rolls will be between $0.5$ give-or-take two standard deviations, or between:

• $0.5-(2\times 0.1)=0.3$ and
• $0.5+(2\times 0.1)=0.7$.

This interval tell us what values of $\hat{p}$ are likely to be observed in samples of size 25. Most of the time (i.e. approximately 95% of the time), the value of $\hat{p}$ is expected to be between 0.30 and 0.70. (For instance, in the animation above, all ten sets of 25 rolls (or 100%) had a sample proportion betweeen 0.30 and 0.70.)

More formally, the sample proportion $\hat{p}$ is likely to lie within the interval

$$p\pm (\text{multiplier}\times \text{s.e.}(\hat{p}\left)\right),$$ where $\text{s.e.}\left(\hat{p}\right)$ is the standard error of the sample proportion (calculated using Eq. (20.1)). The symbol ‘$\pm$’ means ‘plus or minus,’ or ‘give-or-take.’

The multiplier depends on how confident we wish to be that the interval contains the value of $\hat{p}$.

For a 95% interval—the most common level of confidence—the multiplier is approximately 2, based on the 68–95–99.7 rule: Approximately 95% of observations are within two standard deviations of the value of $p$ (the mean of the normal distribution in Fig. 20.1).

That is, the approximate 95% interval is:

$$p\pm (2\times \text{s.e.}(\hat{p}\left)\right).$$ For a 90% interval, either tables or a computer would be used to find the correct multiplier, since the 68–95–99.7 rule isn’t helpful.

In practice, 95% intervals are the most common, and we’ll use a multiplier of $2$ to find an approximate 95% interval when computing the interval without using software. Software can be used for any other percentage interval (or for an exact 95% interval).

In general, higher confidence means wider intervals (Fig. 20.2), since wider intervals are needed to be more certain that the interval contains $\hat{p}$.

FIGURE 20.2: To have greater confidence that the interval will include the sample proportion, the interval needs to be wider