A confidence interval provides a range of plausible values for an unknown population parameter, constructed from sample data. A 95% confidence interval, for example, does not mean there is a 95% probability that the specific interval contains the parameter. Rather, it means that if we repeated the sampling procedure many times and constructed an interval from each sample, about 95% of those intervals would capture the true parameter. This repeated-sampling interpretation is the correct way to read a confidence level.
The general structure of a confidence interval is the point estimate plus or minus a margin of error. For a population mean when the standard deviation \(\sigma\) is known or the sample is large, the interval is \(\bar{x} \pm z^* \cdot \sigma / \sqrt{n}\), where \(z^*\) is the critical value from the standard normal distribution (1.96 for 95%, 2.576 for 99%). When \(\sigma\) is unknown and must be estimated by the sample standard deviation s, the t-distribution is used instead, giving \(\bar{x} \pm t^* \cdot s / \sqrt{n}\), where \(t^*\) has \(n - 1\) degrees of freedom. For large n, the t critical value approaches the corresponding z value. For a population proportion, the interval is \(\hat{p} \pm z^* \cdot \sqrt{\hat{p}(1 - \hat{p}) / n}\), valid when \(n\hat{p} \geq 10\) and \(n(1 - \hat{p}) \geq 10\).
The margin of error \(E = z^* \cdot \sigma / \sqrt{n}\) represents the maximum expected difference between the sample estimate and the true parameter at the given confidence level. Several factors influence the width of an interval. Increasing the sample size decreases the standard error and narrows the interval, providing more precision. Increasing the confidence level, for example from 95% to 99%, widens the interval because a larger critical value is required, illustrating the trade-off between confidence and precision. There is no free lunch: higher confidence means a wider, less precise interval, while lower confidence yields a narrower but less reliable one.