Mathfolis

Significance Test for One Mean

Unit 7 · Inference for Quantitative Data: Means

What AP Stats asks here

A one-mean t-test asks whether xˉ\bar{x} is far enough from a claimed μ0\mu_0 to reject the null. Use t (not z) because σ\sigma is replaced by the sample SD ss. AP rubrics reward correct hypothesis framing, the right df, an honest p-value interpretation, and a conclusion in context — never 'accept H0H_0'.

Test statistic

t-statistic
t=xˉ−μ0s/nt = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}
Degrees of freedom
df=n−1\text{df} = n - 1

Hypothesis framing

Two-sided
Ha:μ≠μ0H_a: \mu \neq \mu_0
Right-tailed
Ha:μ>μ0H_a: \mu > \mu_0
Left-tailed
Ha:μ<μ0H_a: \mu < \mu_0
AP Tip: Use t whenever σ is unknown — essentially always. The z-test for a mean requires a known population SD and is rarely applicable in practice.
Caution: Choose one- vs two-sided BEFORE seeing the data. Switching to one-sided after peeking is p-hacking and inflates the Type I rate.
Type 1

Set up and compute t

Translate the wording into one- or two-sided hypotheses, then compute t with df =n−1= n - 1.

Example 1
A factory claims bolts have μ=5.00\mu = 5.00 cm. An SRS of 36 bolts gives xˉ=5.03\bar{x} = 5.03 cm and s=0.08s = 0.08 cm. For a two-sided test at α=0.05\alpha = 0.05: (A) t≈2.25t \approx 2.25, p-value ≈ 0.031, reject — evidence the mean differs from 5.00. (B) t≈2.25t \approx 2.25, p-value ≈ 0.015, reject. (C) t≈2.25t \approx 2.25, p-value ≈ 0.062, fail to reject. (D) z≈2.25z \approx 2.25, p-value ≈ 0.025, reject.

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Type 2

Why t (not z)

Using ss in place of σ\sigma adds uncertainty. The t-distribution has heavier tails to compensate — especially for small df.

Example 2
A student plugs the sample SD into a z-test formula and reads off the normal-distribution p-value. Which critique is correct? (A) The student is correct — z and t are interchangeable for sample SDs. (B) When σ\sigma is unknown, ss adds uncertainty; use t with df =n−1= n - 1. The z-test understates the variability of xˉ\bar{x}, so the resulting interval is too narrow and the test over-rejects. (C) Z is always wrong in inference. (D) The difference only matters when n>100n > 100.

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Type 3

Two-sided test and CI agreement

A two-sided test of H0:μ=μ0H_0: \mu = \mu_0 at level α\alpha rejects iff the (1−α)(1-\alpha) CI excludes μ0\mu_0.

Example 3
A 95% CI for mean screen time is (5.545,6.855)(5.545, 6.855) hours. The school claims μ=5.5\mu = 5.5 hours. For a two-sided test at α=0.05\alpha = 0.05: (A) Fail to reject — 5.5 is plausible. (B) Reject — 5.5 is just outside the lower bound of the 95% CI, so the test at α=0.05\alpha = 0.05 rejects H0:μ=5.5H_0: \mu = 5.5. (C) Cannot tell — the test and the CI are unrelated. (D) Reject only at α=0.10\alpha = 0.10.

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