21.1 General comments
The previous chapter discussed forming a confidence interval (CI) for one proportion. We will also study CIs in other contexts too.
The following applies to all CIs:
- CIs are formed for the unknown population parameter (such as the population proportion \(p\)), based on a sample statistic (such as the sample proportion \(\hat{p}\)).
- CIs give an interval in which the sample statistic is likely to lie, over repeated sampling.
- Loosely speaking, this is usually interpreted as the CIs giving an interval which is likely to straddle the value of the unknown population quantity. That is, the CI gives an interval of plausible values of the population parameter that may have produced the observed sample statistic.
- Most CIs have the form \[ \text{Statistic} \pm \overbrace{(\text{Multiplier} \times \text{standard error})}^{\text{Called the `margin of error'}}. \]
- The multiplier is approximately 2 for a 95% CI (from the 68–95–99.7 rule).
- The margin of error is \((\text{Multiplier} \times \text{standard error})\).
- The statistical conditions should always be checked to see if the CI is (at least approximately) statistically valid.