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What Is a Confidence Interval?
A confidence interval is a range of values that is likely to contain the true population parameter. A 95% confidence interval means that if you repeated the sampling process many times, about 95% of the intervals would contain the true population mean. It quantifies the uncertainty in your estimate.
How Confidence Intervals Are Calculated
The formula is: CI = x̄ ± z × (σ / √n), where x̄ is the sample mean, z is the z-score for the chosen confidence level, σ is the standard deviation, and n is the sample size. The margin of error is z × (σ / √n). Larger samples and lower confidence levels produce narrower intervals.
Choosing a Confidence Level
Common confidence levels are 90%, 95%, and 99%. A 95% level is the most widely used in research. Higher confidence levels give wider intervals (more certainty but less precision). The choice depends on the consequences of being wrong and the cost of wider intervals.
Common Misconceptions
A 95% confidence interval does NOT mean there is a 95% probability the true mean is in the interval. The true mean is either in the interval or it is not. The 95% refers to the long-run frequency of intervals that capture the true mean across repeated sampling.
Frequently Asked Questions
Larger sample sizes produce narrower confidence intervals because the standard error (σ/√n) decreases as n increases. Quadrupling the sample size cuts the margin of error in half.
The margin of error is the "± " part of the confidence interval. It equals z × (σ / √n) and represents the maximum expected difference between the sample mean and the true population mean at the chosen confidence level.
Use the t-distribution when sample size is small (typically n < 30) and the population standard deviation is unknown. For large samples, the t and z distributions are nearly identical.
Yes. If your sample mean is close to zero and the margin of error is large, the lower bound can be negative. This is valid and simply reflects the uncertainty in the estimate.