A confidence interval is a fundamental concept in statistics. It provides a range of values that likely contains an unknown population parameter. The interval consists of a sample statistic, such as the sample mean, plus or minus a margin of error. This creates a range with a specified confidence level, typically ninety-five percent.
The confidence level tells us about the long-run behavior of our method. A ninety-five percent confidence level means that if we repeated our sampling process many times, about ninety-five percent of the intervals would contain the true population parameter. However, we cannot know whether our specific interval contains the true value or not.
To calculate a confidence interval, we use the formula: confidence interval equals sample mean plus or minus the critical value times the standard error. The critical value depends on our confidence level, and the standard error is the population standard deviation divided by the square root of sample size. This creates our margin of error around the sample mean.
The width of a confidence interval depends on three key factors. First, the confidence level: higher confidence requires a wider interval. Second, the sample size: larger samples give narrower intervals. Third, population variability: more variable populations require wider intervals. We can see these relationships in the formula for interval width.
To summarize what we have learned about confidence intervals: They provide estimated ranges for unknown population parameters with specified uncertainty levels. The confidence level tells us about the long-run success rate of our method. Interval width is affected by confidence level, sample size, and population variability. These intervals are essential tools for making statistical inferences and informed decisions based on sample data.