A confidence interval is a range of values that is likely to contain the true population parameter (mean or proportion) with a certain level of confidence. For example, a 95% confidence interval means that if you repeated the sampling process many times, approximately 95% of the calculated intervals would contain the true population value.

Use the Z-distribution when the population standard deviation (σ) is known, or when the sample size is large (n ≥ 30). Use the T-distribution when σ is unknown and estimated from the sample, especially for small sample sizes (n < 30). The T-distribution has heavier tails to account for the extra uncertainty from estimating σ.

The Wilson score interval is a more accurate method for calculating confidence intervals for proportions, especially when sample sizes are small or the proportion is near 0 or 1. Unlike the traditional Wald interval, the Wilson interval never produces impossible bounds (below 0 or above 1) and has better actual coverage probability.

Larger sample sizes produce narrower confidence intervals because the standard error decreases as n increases (SE = s/√n for means). Doubling the sample size reduces the interval width by approximately 29% (1/√2 ≈ 0.707). To cut the interval width in half, you need to quadruple the sample size.

The margin of error (ME) is the half-width of the confidence interval. It equals the critical value multiplied by the standard error (ME = z* × SE or t* × SE). A smaller margin of error indicates a more precise estimate. The margin of error decreases with larger sample sizes and increases with higher confidence levels.

The 95% confidence level is widely used because it provides a good balance between precision (narrow interval) and reliability (high probability of capturing the true value). A 99% interval is wider (less precise) but more reliable, while a 90% interval is narrower (more precise) but less reliable. The 95% level has become a standard convention in many scientific fields.

Yes, but with caution. For very small samples (n < 10), the T-distribution with n-1 degrees of freedom is essential. The resulting interval will be quite wide, reflecting the high uncertainty. Always check that your data approximately follows a normal distribution, as the T-distribution assumption relies on normality for very small samples.

A confidence interval estimates where the population mean lies. A prediction interval estimates where a single future individual observation will fall. Prediction intervals are always wider than confidence intervals because they account for both the uncertainty in estimating the mean and the natural variability of individual data points.