| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Explain Type I or II error |
| Difficulty | Moderate -0.3 This is a straightforward application of Type I error definition to a standard binomial hypothesis test. Part (i) requires recalling the definition in context, and part (ii) involves a routine binomial probability calculation P(X ≤ 1 | p=0.2, n=15). Both parts are textbook exercises with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) George says there are fewer than 20% red chocolate beans when there are 20% | B1 (1 mark) | Or equivalent, relating to the question |
| (ii) \(P(X = 0 \text{ or } 1) = 0.8^{15} + 0.8^{14} \times 0.2 \times {}_{15}C_1 = 0.167\) | B1, B1, B1 (3 marks) | For identifying correct outcome; For correct unsimplified expression; For correct answer |
**(i)** George says there are fewer than 20% red chocolate beans when there are 20% | B1 (1 mark) | Or equivalent, relating to the question
**(ii)** $P(X = 0 \text{ or } 1) = 0.8^{15} + 0.8^{14} \times 0.2 \times {}_{15}C_1 = 0.167$ | B1, B1, B1 (3 marks) | For identifying correct outcome; For correct unsimplified expression; For correct answer
---2 A manufacturer claims that $20 \%$ of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than $20 \%$ and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis $p = 0.2$ against the alternative hypothesis $p < 0.2$. He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.\\
(i) With reference to this situation, explain what is meant by a Type I error.\\
(ii) Find the probability of a Type I error in George's test.
\hfill \mbox{\textit{CAIE S2 2005 Q2 [4]}}