The CI and the two-sided test agree

Two views of one calculation

A two-sided test at level $\alpha$ rejects $H_0:\mu ={\mu }_0$ when

$$\left|\frac{\bar{x}-{\mu }_0}{\mathrm{SE}}\right|>t_{\alpha /2},\text{i.e.}|\bar{x}-{\mu }_0|>t_{\alpha /2}\times \mathrm{SE}$$

The right-hand side is the margin of error of the $(1-\alpha )$ confidence interval. So the test rejects exactly when ${\mu }_0$ is further from $\bar{x}$ than the margin — exactly when ${\mu }_0$ lies outside the interval. The duality:

> ${\mu }_0$ is outside the $(1-\alpha )$ confidence interval $⟺$ the two-sided test rejects $H_0:\mu ={\mu }_0$ at level $\alpha$.

A confidence interval is the set of all null values the data would not reject. L9.2's consultant-claim question was this in disguise.

Worked example — is the mean audit fee \$200?

L9.4 built Cedar & Stone's 95 % interval for the mean audit fee from $n=10$, $\bar{x}=212.4$, $s=18.6$: $(199.10,225.70)$. The managing partner asks whether the mean fee differs from the \$200 the firm advertises. Two-sided: $H_0:\mu =200$, $H_a:\mu \ne 200$, at $\alpha =0.05$.

By the interval. 200 lies inside $(199.10,225.70)$ — just. Do not reject at 5 %: there is insufficient evidence at the 5 % level that the mean fee differs from \$200.

By the statistic. $\mathrm{SE}=18.6/\sqrt{10}=5.882$;

$$t=\frac{212.4-200}{5.882}=2.108,t_{0.025;9}=2.262$$

$2.108<2.262$: do not reject. The two routes agree, as the algebra says they must: $2.108<2.262$ is the same inequality as $12.4<13.30$, the margin.

Had the advertised fee been \$195, outside the interval, the test would reject — and $t=17.4/5.882=2.958>2.262$ confirms it. The interval answers every two-sided test at once: any ${\mu }_0$ below 199.10 or above 225.70 is rejected at 5 %, any ${\mu }_0$ between is not.

Why the interval is often the better answer

The test says whether 200 is plausible; the interval says which values are — 199.10 to 225.70 — and shows that 200 survives only by 90 cents. A manager reading the interval sees at once that the mean fee is probably above \$200 and that the sample is too small to be sure. The test alone hides both facts behind "do not reject". When the paper asks for both, do both; when it asks for one, the interval usually says more.

Levels and tails must match

The duality pairs a $(1-\alpha )$ interval with a level-$\alpha$ two-sided test: 95 % with $\alpha =0.05$, 99 % with $\alpha =0.01$. A 95 % interval says nothing directly about a test at 1 %. And a one-sided test pairs with a one-sided interval (a bound, open on one side — R prints one in L10.8) — not with the ordinary two-sided interval. This course names the one-sided interval and does not build it.