IntroStatsCalc

One-Sample T-Test

Hypothesis test for a mean when σ is unknown, from data or summary statistics.

Inputs

Enter
Alternative hypothesis Ha
Significance level α

Results update as you type.

Results

Reject H₀: t = 1.8454, p-value = 0.0396

The p-value (0.0396) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the population mean is greater than 30.

With n = 22, this test assumes the population is approximately normal.

t-distribution with 21 degrees of freedom curve. The shaded p-value region is the area to the right of t = 1.85, with area 0.0396. Dashed line marks the critical value 1.721.−4−2024t = 1.85critical 1.721

Results

Test statistic t1.8454
p-value0.0396
df21
x̄32.4
n22
Standard error1.3005
Critical value(s)1.721
Decision at α = 0.05Reject H₀

On the TI-84

STAT → TESTS → 2:T-Test. Choose Data or Stats, enter μ0, x̄, Sx, n, pick the alternative, Calculate (or Draw).

T-Test 30, 32.4, 6.1, 22, >

Show steps
  1. Hypotheses. H₀: μ = 30   Ha: μ > 30
  2. Test statistic. t = (x̄ − μ0) ÷ (s ÷ √n) = (32.4 − 30) ÷ (6.1 ÷ √22) = 2.4 ÷ 1.3005 = 1.8454, with df = 21
  3. p-value. Right-tailed test: area to the right of the test statistic: p-value = 0.0396. Since 0.0396 ≤ 0.05, reject H₀.
  4. Critical-value method. At α = 0.05, the rejection region is t ≥ 1.721. The test statistic t = 1.8454 is in the rejection region, so we reject H₀. (Same decision as the p-value method.)
  5. Conclusion. The p-value (0.0396) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the population mean is greater than 30.

About this calculator

When to use a t-test

This is the usual test for one mean: you have a sample, its mean x̄ and standard deviation s, and a claimed value μ0. The test statistic follows a t-distribution with df = n − 1.

Conditions

The data should come from a random sample. If n is less than about 30, the population should be roughly normal: make a histogram or boxplot and look for strong skew or outliers.

Writing the conclusion

Always answer in context. If you reject H0, there is enough evidence for the alternative claim. If you fail to reject, there isn't enough evidence, which is not the same as proving H0 is true.

More inference calculators: CI for a Mean · CI for a Proportion · Sample Size · Z-Test (mean) · 1-Prop Z-Test · 2-Sample T-Test · Paired T-Test · 2-Prop Z-Test