Hypothesis tests for a binomial proportionEdexcel A-Level Maths: Revision notes

To test a claim about a proportion, define the parameter $p$ (for example, the proportion of all parcels delivered on time), then write hypotheses in terms of $p$: $H_0:p=p_0\qquad H_1:p>p_0,\;p<p_0\text{ or }p\ne p_0.$ The sample is used to make an inference about the population. If $H_0$ is true, the number of successes in the sample is $X\sim B(n,p_0)$. The significance level is the probability of incorrectly rejecting $H_0$ when it is true. (A formal treatment of Type I errors is not required.)

1. Define $p$ and write $H_0$ and $H_1$.
2. State the distribution of $X$ under $H_0$.
3. Find the $p$-value: for $H_1:p>p_0$ it is $P(X\ge x)$; for $H_1:p<p_0$ it is $P(X\le x)$, where $x$ is the observed value.
4. Compare with the significance level and conclude in context. Example: a drug cures 30% of patients. 10 of 20 patients on a new drug are cured. $X\sim B(20,0.3)$, $H_1:p>0.3$. $p$-value $=P(X\ge10)=1-P(X\le9)=0.0480<0.05$, so reject $H_0$: there is evidence that the new drug cures a greater proportion.

If $H_1:p\ne p_0$, the significance level is split equally: compare the tail probability with half the significance level. Decide which tail from the expected value $np_0$. For $X\sim B(30,0.4)$ the mean is 12, so an observation of 17 is in the upper tail and the $p$-value is $P(X\ge17)=0.0481$. At the 5% level compare with $0.025$: $0.0481>0.025$ so do not reject $H_0$. Alternatively, double the one-tail probability and compare with the full level.

Instead of a $p$-value, find the critical region first, then see whether the observation lies in it. Example: $X\sim B(40,0.05)$, $H_1:p>0.05$, 1% level. $P(X\ge6)=0.0139>0.01$ but $P(X\ge7)=0.0034\le0.01$, so the critical region is $X\ge7$ and the actual significance level is $0.0034$. An observation of 6 is not in the critical region, so do not reject $H_0$. Choose the critical region as large as possible with probability at most the significance level.

A full conclusion has two parts: a statement about $H_0$ and a statement in context.

• Reject $H_0$: 'There is evidence at the 5% level that the new drug cures more than 30% of patients.'
• Do not reject $H_0$: 'There is insufficient evidence at the 5% level that the proportion of parcels delivered on time is less than 85%.' Failing to reject does not prove the claim; the sample may simply be too small to show a difference. The test shows evidence, never certainty.