Confidence Interval

A confidence interval (CI) is a range of values that is likely to include an unknown population parameter. The interval is constructed using an estimate of the parameter and a measure of uncertainty. The uncertainty is often expressed as a standard deviation or a confidence level.

The confidence level is the probability that the true value of the parameter lies within the confidence interval. For example, a 95% confidence interval means that there is a 95% probability that the true value of the parameter lies within the interval.

The width of the confidence interval is determined by the sample size and the variability of the data. The larger the sample size, the narrower the confidence interval. The more variable the data, the wider the confidence interval.

Confidence intervals are used to make inferences about the population from a sample. They can be used to test hypotheses, estimate population parameters, and compare groups.

Here is an example of how a confidence interval can be used to test a hypothesis. Suppose you want to test the hypothesis that the average height of women in the United States is 5 feet 4 inches. You take a random sample of 100 women and find that the average height is 5 feet 5 inches. You can then construct a 95% confidence interval for the true average height of women in the United States. The interval will be something like 5 feet 4 inches to 5 feet 6 inches. This means that there is a 95% probability that the true average height of women in the United States is between 5 feet 4 inches and 5 feet 6 inches.

Confidence intervals are a valuable tool for statistical inference. They allow us to make inferences about the population from a sample, and they can be used to test hypotheses and estimate population parameters.