Binomial hypothesis tests

The idea

A hypothesis test asks whether data give enough evidence to reject $H_0$ . Assume $H_0$ is true, find the probability of a result at least as extreme as the one seen, and compare it with the significance level.

Write $H_0: p = \dots$ and $H_1: p > \dots$ , $p < \dots$ or $p \ne \dots$ .
Find $P(X \ge x)$ or $P(X \le x)$ under $H_0$ . For a two-tailed test, halve the level for each tail.
Compare with the level and conclude in context.

Watch out Not rejecting $H_0$ is not proof that $H_0$ is true.

See it worked

Worked example

$H_0: p = 0.25$ , $H_1: p > 0.25$ , $n = 20$ , and $9$ successes are seen. Test at the $5\%$ level.

Under $H_0$ , $X \sim B(20, 0.25)$ .
$P(X \ge 9) = 1 - P(X \le 8) = 0.0409$ .
$0.0409 < 0.05$ : reject $H_0$ . There is evidence that $p > 0.25$ .

Answer Reject $H_0$ : evidence that $p > 0.25$

Your turn

Question 1Calculator allowed3 marks

$X \sim B(20, 0.3)$ under $H_0$ . For $H_1: p < 0.3$ at the $5\%$ level, the critical region is $X \le c$ . Find $c$ .

Question 2Calculator allowed3 marks

$H_0: p = 0.4$ , $H_1: p > 0.4$ . In $15$ trials there are $10$ successes. Find the $p$ -value to 4 decimal places.

Question 3Non-calculator3 marks

In the test above ( $p$ -value $0.0338$ ), state the conclusion at the $5\%$ level in context: the probability $p$ that a seed germinates.

Write your answer, then show the working

$0.0338 < 0.05$ , so the result is significant.
Reject $H_0$ .
There is evidence that the germination rate is higher than $0.4$ .

Answer Reject $H_0$ : there is evidence that more than $40\%$ of seeds germinate

In past papers

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