What Is a Confidence Interval?
A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval means that if we were to repeat the sampling process many times, approximately 95% of the calculated intervals would contain the true population mean.
Confidence intervals are essential tools in inferential statistics because they quantify the uncertainty in our estimate. Rather than providing a single point estimate, they give a range that accounts for sampling variability, making them more informative for decision-making.
The Formula
Where x̄ is the sample mean, z* is the critical value for the desired confidence level, s is the sample standard deviation, and n is the sample size. The term s/√n is called the standard error of the mean.
Z-Values Table
| Confidence Level | Z-Value | Alpha |
|---|---|---|
| 90% | 1.645 | 0.10 |
| 95% | 1.960 | 0.05 |
| 99% | 2.576 | 0.01 |
Interpretation
A common misinterpretation is that "there is a 95% probability the true mean falls within this interval." The correct interpretation is that 95% of intervals constructed this way from repeated samples would contain the true mean. The actual interval either contains the parameter or it does not.
Wider intervals indicate more uncertainty, while narrower intervals indicate more precision. You can narrow a confidence interval by increasing sample size, decreasing the confidence level, or reducing variability in the data.
Frequently Asked Questions
When should I use a t-distribution instead of z?
Use the t-distribution when the sample size is small (typically n < 30) and the population standard deviation is unknown. For large samples, the t-distribution closely approximates the normal distribution, so z-values work well.
How does sample size affect the interval?
Larger sample sizes produce narrower confidence intervals because the standard error (s/√n) decreases as n increases. Quadrupling the sample size halves the margin of error.
Can I use this for proportions?
For proportions, use the formula CI = p̂ ± z*√(p̂(1-p̂)/n). This calculator is designed for means with known or estimated standard deviations.