Unit 7: Inference for Quantitative Data: Means

Significance Test for a Mean

Conducting a one-sample t-test for a population mean.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Confidence Interval for a Mean

Testing a Claimed Mean with Limited Information

A manufacturer claims their light bulbs last on average 1000 hours. A quality inspector samples 16 bulbs and finds x̄ = 985 hours, s = 40 hours. Before reading on: is a 15-hour shortfall on a sample of just 16 bulbs surprising enough to doubt the manufacturer's claim, or could ordinary sampling variability easily explain it?

Definition — One-Sample t-Test for a Mean

The test statistic is t = (x̄ − μ₀)/(s/√n), with df = n−1, compared against the t-distribution to find a p-value. As with the confidence interval, this requires a random sample, the 10% condition, and either an approximately Normal population or a large enough sample with no strong skew or outliers.

t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}

Worked Example — Conducting and Interpreting a Two-Sided t-Test

H₀: μ = 1000. Ha: μ ≠ 1000. Conditions: random sample (given), 10% condition (large production batch), and since n=16 is small, the sample is assumed checked and reasonably symmetric with no strong skew or outliers. SE = 40/√16 = 10. t = (985−1000)/10 = −1.5, with df = 15. A standard t-table for df=15 lists critical values 1.341 (upper-tail area 0.10) and 1.753 (upper-tail area 0.05) — since |t| = 1.5 falls between them, the one-sided tail area is between 0.05 and 0.10, so the two-sided p-value falls between 0.10 and 0.20.

Since even the smallest value in that range (0.10) already exceeds α = 0.05, fail to reject H₀ regardless of exactly where the true p-value falls within that bound: there is not convincing evidence that the true mean bulb lifetime differs from 1000 hours. This does not prove the true mean is exactly 1000 — it only means this particular sample of 16 bulbs doesn't provide strong enough evidence to conclude otherwise, at the 5% significance level. A larger sample, or a different result, could still reveal a genuine difference.

Tip

A t-table only lists specific critical values, not every possible p-value — bounding a p-value between the two nearest table entries (as done here) is a legitimate, standard technique, especially when exact software output isn't available.

Common Mistakes

Treating 'fail to reject H₀' as equivalent to 'the true mean is 1000 hours.'
A test that fails to reject H₀ has only shown insufficient evidence against it — it never proves the null hypothesis is correct, only that this sample doesn't contradict it strongly enough.

Key Takeaways

A one-sample t-test for a mean uses t = (x̄−μ₀)/(s/√n) with df = n−1.
A t-table can bound a p-value between two known critical values when exact software output isn't used — a standard, legitimate technique.
Failing to reject H₀ means insufficient evidence against the claim, never proof the claim is true.

Summary

This closes Unit 7: inference for means parallels inference for proportions, using the t-distribution in place of the Normal. Unit 8 returns to categorical data, testing distributions and associations across entire tables at once.

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