Confidence Interval Calculator

How to calculate a Confidence Interval?

A confidence interval (CI) provides a range of values that is likely to include an unknown population parameter. While a sample statistic (like the sample mean, x̄) gives us a single point estimate, the confidence interval accounts for the uncertainty and variability inherent in sampling to estimate the true population mean with a specific level of certainty.

CI = x̄ ± Z * (σ / √n)

Using the Confidence Interval calculator: an example

Suppose a researcher measures the IQ scores of a sample of 50 students. The sample mean score is 100, and the known population standard deviation is 15. The researcher wants to calculate a 95% confidence interval for the population mean IQ.

Step-by-step calculation:

1. Determine the Z-score: For a 95% confidence level, the standard Z-score is approximately 1.96.
2. Calculate the standard error: Divide the standard deviation by the square root of the sample size (15 / √50 ≈ 2.121).
3. Compute the margin of error: Multiply the Z-score by the standard error (1.96 * 2.121 ≈ 4.157).
4. Establish bounds: Subtract the margin of error from the mean for the lower bound (95.84) and add it for the upper bound (104.16).

Frequently Asked Questions

How does sample size impact the interval?
Increasing the sample size reduces the standard error, which in turn narrows the confidence interval. A larger sample provides a more precise estimate of the population parameter.

What does "95% confidence" actually mean?
It means that if you were to take 100 different samples and compute a confidence interval for each, approximately 95 of those intervals would contain the true population mean. It describes the reliability of the estimation process.

What is the difference between Z-tests and T-tests?
A Z-test is used when the population standard deviation is known and the sample size is large (typically n > 30). A T-test is used when the population standard deviation is unknown and the sample standard deviation is used as an estimate instead.