Carrying Out a Test for a Mean
Find the p-value
- Convert the $t$ statistic to a p-value using the $t$-distribution at $df = n-1$.
- One-sided $H_a$: the tail area beyond $t$ in that direction.
- Two-sided $H_a$: double the one-tail area.
- The p-value is the chance of a $\bar{x}$ this far from $\mu_0$ if $H_0$ were true.
Compare to alpha
- Compare the p-value to the significance level $\alpha$ (usually $0.05$).
- p $\le \alpha$ → the result is statistically significant.
- p $> \alpha$ → not significant.
- Set $\alpha$ before running the test.
Reject or fail to reject
- p $\le \alpha$ → reject $H_0$ (strong enough evidence against it).
- p $> \alpha$ → fail to reject $H_0$ (not enough evidence).
- Never "accept $H_0$" — only "fail to reject."
- Same decision rule as any significance test.
Conclude in context
- Write the conclusion about the population mean, in context, with units.
- "There is convincing evidence the mean fill is below $500$ mL" (if rejected).
- Or "there is not convincing evidence..." (if not).
- Failing to reject does not prove $H_0$ true.
Get the p-value from the $t$-distribution, not the normal — the correct $df = n-1$ matters, especially for small $n$ where $t$ and $z$ differ most. And match the tail(s) to $H_a$: one tail for one-sided, doubled for two-sided. As always, a large p-value means "insufficient evidence," never "proof of $H_0$."
From 7.4: $t = -2$ with $df = 15$, one-sided $H_a: \mu < 500$, $\alpha = 0.05$.
- p-value: area left of $t = -2$ at $df = 15 \approx 0.032$.
- Compare: $0.032 \le 0.05$ → reject $H_0$.
- Conclude: convincing evidence the true mean fill is below $500$ mL.
Find the p-value from the $t$ statistic at $df = n-1$ (doubled for two-sided), compare to $\alpha$: reject $H_0$ if p $\le \alpha$, else fail to reject. State the conclusion about the population mean in context — and remember failing to reject never proves $H_0$.
The t-test tail area
Read the p-value as a t tail area at df = n − 1.
With p-value = 0.032 and α = 0.05, the decision is...
0.032 ≤ 0.05 → reject H0.
The p-value for a t-test is read from...
Use the t-distribution with the correct df.
For a two-sided mean test, you double the one-tail area to get the p-value.
Both directions are extreme, so double.
Failing to reject H0 proves the population mean equals μ0.
It only means insufficient evidence against H0.
The conclusion of a mean test should be stated...
Tie the conclusion to μ, in context, with units.