Hypothesis test for a binomial proportionAQA A-Level Maths: Revision notes

A test uses a sample to make an inference about the population: the value of $p$ for all trials. Follow the same steps each time.

1. Define $p$ in words and write $H_0$ and $H_1$.
2. State the distribution of the test statistic under $H_0$, such as $X\sim B(12,0.25)$.
3. Use either the $p$-value or a critical region at the stated significance level.
4. Make a decision about $H_0$, then state the conclusion in context. The significance level is the probability of incorrectly rejecting $H_0$ (rejecting it when it is true).

Calculate the probability, under $H_0$, of a result at least as extreme as the observed value $x$ in the direction of $H_1$.

• $H_1:p<p_0$: $p$-value $=P(X\le x)$.
• $H_1:p>p_0$: $p$-value $=P(X\ge x)=1-P(X\le x-1)$. Compare with the significance level. If $p$-value $\le\alpha$, reject $H_0$. Example: historically 25% of buses are late. In 12 journeys 1 is late. $H_0:p=0.25$, $H_1:p<0.25$. $p$-value $=P(X\le1)=0.1584>0.05$, so do not reject $H_0$: there is insufficient evidence that the proportion of late buses has decreased.

Find the critical region for the significance level first, then see whether the observed value lies in it. Example: $H_0:p=0.7$, $H_1:p>0.7$, $X\sim B(20,0.7)$, 5% level. $P(X\ge17)=0.107>0.05$ but $P(X\ge18)=0.0355\le0.05$, so the critical region is $X\ge18$. An observed value of 17 is not in it, so $H_0$ is not rejected. The actual significance level here is $0.0355$, which is lower than $5\%$ because $X$ is discrete.

When $H_1:p\ne p_0$, split the significance level equally between the two tails. At the 5% level, compare the relevant tail probability with $0.025$. Example: a coin gives 9 heads in 10 tosses. $H_0:p=0.5$, $H_1:p\ne0.5$. $P(X\ge9)=0.0107<0.025$, so reject $H_0$: there is sufficient evidence that the coin is biased. Use the tail in the direction of the observed result: the lower tail $P(X\le x)$ if $x$ is below the expected value $np_0$, and the upper tail $P(X\ge x)$ if it is above.

The conclusion has two parts: the decision about $H_0$, then a sentence in context. Use cautious wording.

• Reject $H_0$: "There is sufficient evidence, at the 5% level, that the proportion of defective items is higher than 8%."
• Do not reject $H_0$: "There is insufficient evidence that the cure rate is higher." This does not prove $H_0$ true. The result can depend on the significance level: a $p$-value of $0.0293$ rejects $H_0$ at 5% but not at 1%. Because a sample is used, the test can reach the wrong decision. The probability of incorrectly rejecting $H_0$ is the significance level. A larger sample gives a more reliable estimate of $p$.