A-level · Hypothesis testing

Critical regions and actual significance levelExam-style questions

The critical-region method for a one-tailed test: the critical value is the boundary and the critical region is the set of outcomes that would lead to rejecting H0. How to find it (the smallest c with P(X ≥ c) ≤ α for an upper test, the largest c with P(X ≤ c) ≤ α for a lower test), why the discreteness of the binomial means the actual significance level is usually below the nominal α, and how to use the region to conclude.

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
For a test of $H_0: p = 0.25$ against $H_1: p > 0.25$ with $X \sim B(20, 0.25)$ , find the critical region at the $5\%$ significance level.
Level 3 · 2 marksTry it
2
For a test of $H_0: p = 0.5$ against $H_1: p < 0.5$ with $X \sim B(25, 0.5)$ , find the critical region at the $5\%$ significance level.
Level 3 · 2 marksTry it
3
For a test of $H_0: p = 0.4$ against $H_1: p \ne 0.4$ with $X \sim B(20, 0.4)$ , find the critical region at the $5\%$ significance level.
Level 3 · 3 marksTry it
4
A test of $H_0: p = 0.25$ against $H_1: p > 0.25$ with $X \sim B(20, 0.25)$ uses the critical region $X \ge 9$ . Find the actual significance level of the test.
Level 3 · 2 marksTry it
5
A test of $H_0: p = 0.25$ against $H_1: p > 0.25$ with $X \sim B(20, 0.25)$ has critical region $X \ge 9$ . The observed value is $X = 10$ . State, with a reason, the conclusion of the test.
Level 3 · 1 markTry it

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