Tests and confidence intervals
Section of Hypothesis tests, part of Statistical inference
Tests and confidence intervals
| Type | Section |
| Taught in | Hypothesis tests |
| Part | Statistical inference |
A two-sided test at level \alpha and a confidence interval with confidence 1-\alpha built from the same pivot carry the same information. H_0:\theta=\theta_0 is rejected exactly when \theta_0 lies outside the interval. For the one-sample t test,
\Bigl|\frac{\overline x-\mu_0}{s/\sqrt n}\Bigr|<t_{n-1;1-\alpha/2} \iff \mu_0\in\overline x\pm t_{n-1;1-\alpha/2}\,\frac{s}{\sqrt n}.
A one-sided test pairs with a one-sided bound, as in the caffeine example. The interval is usually the more useful report. It shows which parameter values are compatible with the data, and therefore whether a significant effect is large enough to matter.
Compute how often the test statistic would look this extreme if the null hypothesis were true. That probability is the p-value. It is not the probability that the null hypothesis is true, and a small one does not say the effect is large.
Used in
See also
Null and alternative hypotheses · Critical region · p-value · Type I and type II errors · Neyman–Pearson approach · Power function · One-sample t test · Two-sample t test