O: Statistical hypothesis testing: Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.

The idea

This topic covers testing a claim about a population proportion using a binomial model. Set up a null hypothesis (H0: p = p0) and an alternative (H1: p < p0, p > p0 or p ≠ p0). Calculate the probability of the observed result or something more extreme, assuming H0 is true. For a one-tailed test, compare this probability to the significance level α. For a two-tailed test, compare the observed tail probability to α/2. If the probability is less than or equal to the threshold, reject H0. Interpret the decision in context, noting the sample provides evidence about the unknown population proportion. The significance level α is the probability of incorrectly rejecting a true H0.

What earns the marks

• Correctly state H0 and H1 in terms of the population proportion p, ensuring the alternative hypothesis matches the direction of the investigation.
• Identify the binomial distribution X ~ B(n, p0) under the null hypothesis, where n is the sample size and p0 is the hypothesised proportion.
• Calculate the appropriate tail probability for the observed test statistic, comparing it to α for a one-tailed test or α/2 for a two-tailed test.
• Make a formal decision to reject or not reject H0 and write a conclusion in context, linking back to the original claim about the population proportion.
• Explain that the significance level is the probability of incorrectly rejecting a true null hypothesis (a Type I error).

Common mistakes and how to put them right

• Doubling the tail probability for a two-tailed test.

  Put it right: Compare the observed tail probability directly to α/2.

• Using the sample proportion as the null hypothesis proportion.

  Put it right: The null hypothesis must use the claimed population proportion p0, not the observed sample proportion.

• Stating the conclusion as 'accept H0' rather than 'do not reject H0'.

  Put it right: The test does not prove H0, it only fails to find sufficient evidence against it.

• Misinterpreting the significance level as the probability that H0 is true.

  Put it right: It is actually the probability of rejecting H0 given that H0 is true.

Exam tips

• Always define the population proportion p in words before writing H0 and H1, so your hypotheses are unambiguous.
• Show the distribution you are using under H0, e.g. X ~ B(20, 0.3), and state the observed value of X clearly.
• For two-tailed tests, it is often safer to find the critical regions for each tail using α/2 rather than calculating p-values, especially if the distribution is highly skewed.