One-sample t-test
One-sample t-test assesses whether a sample mean is consistent with a hypothesised population value.
Also known assingle-sample t-test
FrameworkStudent's t-test
See it move
The infographic is a formula card showing the one-sample t-statistic t = (x̄ − μ₀) ÷ (s ÷ √n), where x̄ is the sample mean, μ₀ the claimed population mean, s the sample standard deviation, and degrees of freedom equal n − 1. A worked example tests a claim that bags weigh 500 g: with n = 16, x̄ = 496, and s = 8, the statistic is (496 − 500) ÷ (8 ÷ √16) = −2.0, which falls short of the critical value ±2.131 at df = 15 and α = 5%, so the claim is not rejected; the card notes that t-critical values, not z = 1.96, must be used when variance is estimated from the sample.
The formula
Variables
- Test statistic, compared against a t-distribution with n − 1 degrees of freedom
- Sample mean
- Hypothesised population mean under H₀
- Sample standard deviation
- Sample size
The denominator s ÷ √n is the standard error of the mean; a larger sample shrinks the standard error and sharpens the test.
Check yourself
A bottling line is set to fill bottles to a mean of 330 ml. A quality check weighs 25 bottles and finds a sample mean of 327 ml and a sample standard deviation of 5 ml. What is the t-statistic, and how many degrees of freedom does this test use?