Confidence Interval (Mean)

This formula gives a confidence interval for a population mean by taking the sample mean as the center and adding/subtracting a margin of error. The margin of error comes from the standard error, \(\sigma/\sqrt{n}\), which shrinks as the sample size grows because larger samples usually give a more precise estimate of the true mean. The z-value scales that uncertainty to match the chosen confidence level, so a higher confidence level produces a wider interval.

Use this when you want to estimate a population mean and the population standard deviation \(\sigma\) is known, or when a large-sample normal approximation is justified. Common examples include manufacturing quality control, standardized test analysis, and scientific measurements where repeated samples are used to estimate an average.

• Using the sample standard deviation s in this formula when the problem specifically requires known \u03c3
• Forgetting to divide by \u221an, which makes the interval far too wide
• Interpreting the interval as the probability that the fixed population mean lies inside it, rather than as a method that captures the mean in repeated sampling