Hypothesis Tests for the Rate of a Poisson Distribution

Test hypotheses about the rate parameter $\lambda$ of a Poisson distribution using the observed count as the test statistic. Compute upper-tail, lower-tail p-values from the Poisson CDF, and extend to counts collected over a time interval of length $t$ using $\lambda_0 t$ under the null.

Step 1 of 119%

Tutorial

Upper-Tail Test for a Poisson Rate

Suppose $X \sim \text{Poisson}(\lambda)$ for some unknown rate $\lambda > 0$ . To test whether the rate has increased beyond a baseline value $\lambda_0,$ we set up the hypotheses

H_0\!: \lambda = \lambda_0 \qquad \text{vs.} \qquad H_1\!: \lambda > \lambda_0.

The test statistic is the observed count $x.$ Under $H_0,$ this count follows $\text{Poisson}(\lambda_0),$ so the p-value is the upper-tail probability of observing a count at least as extreme as $x{:}$

p = P(X \geq x \mid \lambda = \lambda_0) = 1 - F(x-1;\, \lambda_0),

where $F(k;\,\lambda_0) = P(X \leq k \mid \lambda = \lambda_0)$ is the Poisson CDF. We reject $H_0$ at significance level $\alpha$ when $p \leq \alpha.$

For example, suppose a call center historically receives $\lambda_0 = 3$ calls per hour, and one hour we observe $x = 7$ calls. Given $F(6;\, 3) \approx 0.9665,$

p = P(X \geq 7 \mid \lambda = 3) = 1 - F(6;\,3) = 1 - 0.9665 = 0.0335.

At $\alpha = 0.05,$ since $0.0335 < 0.05,$ we reject $H_0$ and conclude that the rate appears to have increased.