Hypothesis Tests for Binomial and Poisson Distributions (9709 S2 §6.5)

Core Concept

The Hypothesis-Testing Framework

A hypothesis test for a Binomial or Poisson parameter follows five stages:

1. State $H_0$ and $H_1$ — the null hypothesis fixes the parameter; the alternative states the direction of interest.
2. Choose significance level $\alpha$ (e.g. 5%, 1%, 10%) — given in the question.
3. Identify the critical region / compute the $p$-value — using exact probabilities or a normal approximation.
4. Compare — if $p\text{-value} \leq \alpha$, reject $H_0$; otherwise do not reject $H_0$.
5. Conclude in context — a statement referring to the original scenario.

Binomial Tests

If a random variable $X \sim \text{B}(n, p)$ under $H_0$, a single observation $x$ is made. The $p$-value is the probability of obtaining a value at least as extreme as $x$ in the direction of $H_1$:

• One-tailed (upper): $H_1: p > p_0$ → $p\text{-value} = P(X \geq x)$
• One-tailed (lower): $H_1: p < p_0$ → $p\text{-value} = P(X \leq x)$
• Two-tailed: $H_1: p \neq p_0$ → $p\text{-value} = 2 \times \min\bigl(P(X \leq x),\, P(X \geq x)\bigr)$, or find both critical regions and check which tail $x$ falls in.

Poisson Tests

If $X \sim \text{Po}(\lambda)$ under $H_0$, the same logic applies with the Poisson probability formula:

Cumulative Poisson probabilities are evaluated directly (or read from tables if permitted).

Normal Approximation

When $n$ is large and $np > 5$ and $n(1-p) > 5$, approximate $\text{B}(n,p)$ by $\text{N}(np,\, np(1-p))$.
When $\lambda > 15$ (approximately), approximate $\text{Po}(\lambda)$ by $\text{N}(\lambda, \lambda)$.
Apply a continuity correction in both cases.