Confidence interval
Confidence interval: a range derived from sample data that, across repeated samples, would enclose the true population parameter a stated share of the time — 95% of 95% intervals do so.
Also known asCI
See it move
A formula card displays the confidence interval formula x̄ ± t(α/2, n−1) × (s ÷ √n), identifying each component: x̄ as the sample mean, t(α/2, n−1) as the t critical value, s as the sample standard deviation, and n as the sample size. A worked example with 20 students (x̄ = 64, s = 8) yields the interval approximately [60.3, 67.7] using a critical value of 2.093. The visual notes that a larger standard error or smaller sample size widens the interval, and that 95% confidence is a property of the repeated procedure rather than a guarantee for any single computed interval.
The formula
Variables
- sample mean
- critical value from the t-distribution with n − 1 degrees of freedom at two-tailed significance level α
- sample standard deviation
- sample size
Two-sided interval for the population mean when the population standard deviation is unknown. Replace t with z when n is large and σ is known.
Check yourself
A sample of 25 student exam scores has a mean of 68 and a standard deviation of 10. A lecturer constructs a 95% confidence interval for the population mean and states: 'There is a 95% probability that the true mean lies inside this particular interval.' Is she correct?