How to do a t-test

What a one-sample t-test asks

You have a sample mean and want to know whether it is compatible with a claimed population mean μ0 when σ is unknown. Example: “Is the average wait different from 5 minutes?”

Step-by-step

1. Hypotheses. H0: μ = μ0. HA is >, <, or ≠ depending on the wording.
2. Check conditions as your course teaches them (random sample, approximate normality or large n).
3. Compute t = (x̄ − μ0) / (s / √n), df = n − 1.
4. P-value from a t table or software (one tail or two tails to match HA).
5. Decision. If p ≤ α, reject H0; otherwise fail to reject H0. Write the conclusion in context.

Mini example

n = 16, x̄ = 5.4, s = 0.8, μ0 = 5, two-sided, α = 0.05.

1. SE = 0.8 / 4 = 0.2.
2. t = (5.4 − 5) / 0.2 = 2.0, df = 15.
3. Two-sided p-value is about 0.064, which is greater than 0.05.

Useful tips

• “Fail to reject” is not the same as “prove H0 true.”
• A matching 95% confidence interval that contains μ0 agrees with failing to reject a two-sided test at 0.05.
• If the problem is a messy paragraph, scan it after you write H0/HA yourself.

Common mistakes

• Using z and a normal table when the problem gives s, not σ.
• One-tailed p-value for a two-sided claim (or the reverse).
• Reporting t without df.
• Confusing this one-sample test with a two-sample or paired design — those need different formulas. The in-app calculator is a t-test tool as listed; check that your homework matches a one-sample mean if that is what you enter.

How Statistics Problem Solver Pro helps

The listing includes a T-Test calculator among the free offline tools. For the written solution (hypotheses, substitutions, highlighted answer), scan or type the problem. The AI tutor can explain an assignment if you need the “why” after you have the number.

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