Binomial and Poisson hypothesis tests

What you’ll learn

• State hypotheses about a population parameter.
• Calculate an appropriate tail probability or critical region.
• Give a conclusion with its significance and context.

Understand the idea

The null hypothesis specifies a reference value of a population parameter; the alternative gives the change being investigated. Choose a one-sided or two-sided alternative before inspecting the result. Calculate probabilities using the null model, not an estimated parameter substituted from the very observation being tested.

A tail probability measures how unusual the observed result, or a more extreme one in the chosen direction, would be under the null. Reject the null when this probability is at most the chosen significance level. Alternatively, construct a critical region whose null probability does not exceed the allowed level. Discrete counts often make the actual significance smaller than the nominal value.

For a two-sided discrete test, a standard approach allocates at most half the significance to each tail, following any specified convention. A significant result is evidence against the null model, not certainty. A non-significant result means insufficient evidence of change; it does not prove the null hypothesis.

Key formulae

$$H_0:p=p_0,\qquad H_1:p>p_0$$

$$P_{H_0}(X\geq x_{\mathrm{obs}})$$

$$P_{H_0}(X\in C)\leq\alpha$$

$$\text{actual significance}=P_{H_0}(\text{reject }H_0)$$

Worked example

A fair-selection claim says success has probability 0.5. In 12 independent trials there are 10 successes. Test for an increased success probability at the 5% level.

1. State $H_0:p=0.5$ and $H_1:p>0.5$. Under $H_0$, $X\sim B(12,0.5)$.
2. The relevant upper tail is $P(X\geq10)$.
3. $P(X\geq10)$ $=(\binom{12}{10}+\binom{12}{11}+\binom{12}{12})/2^{12}$ $=79/4096$ $=0.01929$.
4. Since $0.01929<0.05$, reject $H_0$. There is evidence of an increased success probability.

Answer: Reject the null at 5%; the data provide evidence that the success probability exceeds 0.5.