The idea

Critical regions and actual significance level

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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Statistics · Hypothesis testing

Critical regions and actual significance level

Why it works

Deciding in advance

A factory tests a batch every hour against the same claim — recomputing a tail probability sixty times a day would be absurd. Instead, work out in advance which outcomes would cause us to reject

H_0

. That set is the critical region (CR), and its boundary value is the critical value. Then the test is just: is the observed value in the critical region?

Finding the critical value

Assume

H_0 : p = p_0

, so

X \sim B(n, p_0)

Upper test ( $H_1 : p > p_0$ ): the CR is $X \ge c$ , where $c$ is the smallest value with $P(X \ge c) \le \alpha$ .
Lower test ( $H_1 : p < p_0$ ): the CR is $X \le c$ , where $c$ is the largest value with $P(X \le c) \le \alpha$ .

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