Mathematics · Hypothesis testing · 5 marks
A manufacturer claims that 20% of its boxes contain a prize. A shopper buys 30 boxes and finds 2 prizes. Test, at the 5% significance level, whether the proportion is lower than claimed. [5]
Mark scheme
- H₀: p = 0.2 · H₁: p < 0.2
- Let X be the number of prizes; under H₀, X ~ B(30, 0.2)
- P(X ≤ 2) = 0.0442 (from the cumulative binomial)
- 0.0442 < 0.05, so the result lies in the critical region
- Reject H₀: there is sufficient evidence at the 5% level that the proportion of boxes with a prize is lower than 20%
A one-tailed test, so the whole 5% sits in the lower tail. The conclusion must be in context and must not claim the manufacturer is lying — only that there is evidence against the claim.