Read the hypothesis test straight off the interval

Question

Someone hands you a 95% confidence interval for a parameter and asks, without any further computation, whether several different null hypotheses would be rejected at the 5% level.

State the duality between confidence intervals and hypothesis tests, and show how to answer a whole family of tests by inspection.

Accepted Answer

The duality. A 95% confidence interval is exactly the set of null values that a two-sided 5% test would fail to reject. Equivalently: 	ext{reject } H0: 	heta = 	heta0 	ext{ at level } \alpha \iff 	heta0 	ext{ lies outside the } (1-\alpha) 	ext{ confidence interval.} So the interval is the acceptance region of the test, re-expressed as a range of parameter values. Every point inside is a null you can't reject; every point outside is a null you can. Using it. Say the 95% CI for a mean effect is [0.24, 3.76]. Without any new arithmetic: - H0: 	heta = 0? Zero is outside → reject at 5%. Th…

Read the hypothesis test straight off the interval

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