How Do You Test a Population Mean Using the Poisson Distribution?

You test a Poisson population mean by stating hypotheses for the Poisson mean $\lambda$, calculating an exact one-tailed probability under $H_0$, and comparing the resulting p-value with the significance level. In IB Mathematics: Applications and Interpretation, this is HL-only content under AHL 4.18.

The Method

Suppose $X$ counts events occurring independently at a constant average rate, so

$$X \sim \operatorname{Po}(\lambda).$$

The parameter $\lambda$ is both the population mean and variance for the specified interval. A Poisson model is appropriate only when events occur independently and at a constant average rate over a fixed interval. If the null mean is stated for a different interval, scale it first. For example, a claimed mean of $3$ events per hour gives $\lambda_0=12$ for a four-hour count. The observed count and $\lambda_0$ must refer to the same interval.

Use these steps:

1. State $H_0$ and $H_1$ in terms of $\lambda$.
2. Assume $H_0$ is true, so .

Worked Example

A call centre claims to receive a mean of $10$ complaints per day. On one day it receives $18$. Test whether the mean has increased at the $5\%$ significance level.

$$H_0:\lambda=10, \qquad H_1:\lambda>10.$$

Under $H_0$, $X\sim\operatorname{Po}(10)$. Therefore,

$$p=P(X\geq18)=1-P(X\leq17)\approx0.0143.$$

Since $0.0143<0.05$, reject $H_0$. There is sufficient evidence at the $5\%$ significance level that the mean number of daily complaints has increased.

Exam Technique

AHL 4.18 restricts Poisson hypothesis tests to one-tailed tests. A common misconception is to say “accept $H_0$”; instead, write “do not reject $H_0$” when $p\geq\alpha$. Examiners also expect the hypotheses, the explicit comparison versus , and a contextual conclusion. Calculator notation may be shown as working, but the final probability should use mathematical notation. Keep full precision until the final comparison.