Critical Regions for Left-Tailed Hypothesis Tests

A left-tailed hypothesis test is one whose alternative claims that the parameter is smaller than the null value. For a Poisson rate, this looks like

$$H_0: \lambda = \lambda_0, \qquad H_1: \lambda < \lambda_0.$$

The critical region (or rejection region) $C$ is the set of values of the test statistic $X$ that lead us to reject $H_0$. For a left-tailed test at significance level $\alpha$, the critical region has the form

$$C = \{0,\, 1,\, 2,\, \ldots,\, c\},$$

where $c$ is the largest non-negative integer satisfying

$$P(X \le c \mid H_0) \le \alpha.$$

We choose the largest such $c$ to make the test as powerful as possible while keeping the probability of a Type I error at most $\alpha$.

For example, suppose $X \sim \text{Poisson}(5)$ under $H_0$ and $\alpha = 0.05$. Using $P(X = k) = e^{-\lambda} \lambda^k / k!$, we compute

$$\begin{align*} P(X \le 0) &= e^{-5} \approx 0.0067, \\ P(X \le 1) &= e^{-5}(1 + 5) \approx 0.0404, \\ P(X \le 2) &= e^{-5}(1 + 5 + 12.5) \approx 0.1247. \end{align*}$$

Since $0.0404 \le 0.05 < 0.1247$, the largest valid value is $c = 1$. The critical region is

$$C = \{0,\, 1\}.$$