Statistics

One-Sample t-Test

Enter your sample statistics (mean, standard deviation, size), the hypothesized population mean μ₀, the significance level α and pick a one- or two-tailed alternative. The calculator computes the t statistic, the standard error, the degrees of freedom, the critical values and the p-value, then concludes whether to reject H₀.

One-Sample t-Test

Test a sample mean against a hypothesized μ₀ — t, df, p-value, conclusion.

Try:
Answert = 1, df = 24, p = 0.327287, fail to reject H₀
  1. Givenx̄ = 52, s = 10, n = 25, μ₀ = 50, α = 0.05
  2. Standard errorSE = s/√n = 10/√25 = 2
  3. Test statistict = (x̄ − μ₀)/SE = 1
  4. Degrees of freedomdf = n − 1 = 24
  5. Critical valueTwo-tailed critical t at α/2 = 0.025: ±2.0639
  6. p-value0.327287
  7. Conclusionp ≥ α — fail to reject H₀: μ = 50 at α = 0.05.

Frequently asked questions

When do I use a t-test instead of a z-test?

Use a t-test when the population standard deviation is unknown and you estimate it from the sample. For large n (≈ 30+) the t and z results converge.

What does 'two-tailed' mean?

Two-tailed tests reject H₀ when the sample mean is significantly larger OR smaller than μ₀. One-tailed tests look only in one direction; they give a smaller p-value for the right side at the cost of missing deviations on the other.

How is the p-value computed?

From the Student-t distribution with df = n − 1, using a regularized incomplete beta function for accuracy across all degrees of freedom.