Confidence Intervals
A confidence interval gives a range of plausible values for an unknown population value, built from sample data. This chapter shows how to construct confidence intervals for a population mean using the normal and the Student t distributions, and for a population proportion, and how the confidence level and sample size affect the interval.
What a Confidence Interval Is
A confidence interval is a range of values, calculated from a sample, that is likely to contain an unknown population value such as a mean or a proportion. It is built as a point estimate plus and minus a margin of error. The confidence level, often ninety-five percent, describes how often intervals built this way would contain the true value if sampling were repeated many times. A wider interval gives more confidence, while a narrower one is more precise.
Intervals for a Mean
When the population standard deviation is known, or the sample is large, a confidence interval for the mean uses the normal distribution, with the margin of error equal to a z-value times the standard error. When the population standard deviation is unknown and estimated from the sample, the Student t distribution is used instead. The t distribution is wider than the normal for small samples and approaches the normal as the sample size grows, which accounts for the extra uncertainty of estimating the standard deviation.
Intervals for a Proportion and Sample Size
A confidence interval for a population proportion is built from the sample proportion plus and minus a margin of error based on the normal distribution. The margin of error depends on the confidence level, the variability in the data, and the sample size. Because a larger sample reduces the margin of error, the required sample size can be calculated in advance from a chosen confidence level and a desired margin of error, letting a study be planned to reach a target precision.