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Significance Tests for a Mean

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The one-sample t test

To test a claim about μ, set H₀: μ = μ₀ against Hₐ (μ <, >, or ≠ μ₀). The t statistic measures how many estimated standard errors x-bar lies from μ₀, and we find the p-value from the t-distribution with n − 1 degrees of freedom. The same conditions as the t-interval apply: Random, 10%, and Normal/Large Sample.

Interval–test duality

A two-sided test at level α and a (1 − α) confidence interval agree: if the hypothesized μ₀ lies outside the confidence interval, you would reject H₀ at level α; if μ₀ lies inside, you fail to reject. This link lets a confidence interval double as a two-sided test, and it is a favorite AP connection to explain.

One-sample t test for a mean
t = (x-bar − μ₀) / ( s / sqrt(n) ) with df = n − 1
Compare the p-value from the t-distribution to α. Reject H₀ if p-value < α; otherwise fail to reject. For a two-sided Hₐ, double the one-tail area.
Worked example

A manufacturer claims mean fill = 500 mL. An SRS of 36 bottles gives x-bar = 496 mL, s = 12 mL. Test Hₐ: μ ≠ 500 at α = 0.05.

  1. 1.Hypotheses: H₀: μ = 500 vs. Hₐ: μ ≠ 500. Conditions: random; n = 36 ≥ 30 so CLT covers shape; < 10% of production.
  2. 2.Standard error: s/sqrt(n) = 12/sqrt(36) = 12/6 = 2.
  3. 3.t = (496 − 500)/2 = −4/2 = −2.0, with df = 35.
  4. 4.Two-sided p-value = 2·P(t₃₅ < −2.0) ≈ 2(0.027) ≈ 0.053, which is greater than 0.05.
Answer: t = −2.0 with df = 35 gives a two-sided p-value ≈ 0.053 > 0.05, so we fail to reject H₀. There is not quite convincing evidence at the 5% level that the mean fill differs from 500 mL.
Checkpoint

A 95% confidence interval for μ is (48, 54). For a two-sided test of H₀: μ = 50 at α = 0.05, what is the decision?

Watch out

A large p-value means "not enough evidence," not "H₀ is true." Failing to reject H₀ that μ = 500 does not prove the mean is exactly 500 — it only means the data are consistent with that value. Never state a conclusion as proof of the null.

Checkpoint

A one-sample t test gives t = 2.8 with df = 20 for a one-sided (upper) alternative. The p-value is the:

Answer the 2 checkpoints as you read.

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