How Do You Test a Population Proportion Using the Binomial Distribution?

Use a binomial hypothesis test by stating hypotheses about the population proportion, calculating the probability of a result at least as extreme as the observed result under the null hypothesis, and comparing this p-value with the significance level. In IB Mathematics: Applications and Interpretation, this is AHL 4.18 and binomial tests are one-tailed.

The Method

Let $p$ be the unknown population proportion, and let $p_0$ be the value claimed under the null hypothesis.

1. State hypotheses using the population parameter $p$, not the sample proportion: $$H_0:p=p_0$$
2. Write the appropriate one-tailed alternative hypothesis.
3. Under $H_0$, define the random variable: $$X\sim B(n,p_0)$$
4. Use a GDC to calculate the relevant tail probability.
5. Compare the p-value with the significance level $\alpha$.
6. Give a conclusion in context.

The model requires a fixed number of independent trials, two possible outcomes per trial, and a constant probability of success. These conditions justify using the binomial model rather than another probability distribution. The null hypothesis supplies the success probability used for every calculation; the sample result determines which tail is evaluated.

Worked Example

A company claims that $50\%$ of customers prefer its new product. In a sample of $20$ customers, $15$ prefer it. Test whether the true proportion is greater than $0.5$ at the $5\%$ significance level.

$$H_0:p=0.5,\qquad H_1:p>0.5$$

Under $H_0$, $X\sim B(20,0.5)$. The GDC gives

$$P(X\geq15)=0.0207.$$

For the upper tail, enter $P(X\geq15)=1-P(X\leq14)$; this includes the observed value and all more extreme outcomes supporting $H_1$.

Since $0.0207<0.05$, reject $H_0$. There is sufficient evidence that more than $50\%$ of customers prefer the new product.

Exam Technique

A common misconception is writing hypotheses using the sample proportion. Examiners expect hypotheses in terms of $p$, the distribution and parameters, an explicit comparison between the p-value and $\alpha$, and a contextual conclusion. Keep full calculator precision until the final comparison. Do not say “accept $H_0$”, say “insufficient evidence to reject $H_0$” when the p-value is larger than .