| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Moderate -0.3 This is a straightforward application of Type I and Type II error definitions with binomial distributions. Part (i) requires calculating P(X≥1) when p=0.0017, part (ii) requires P(X=0) when p=0.0024, and part (iii) uses Poisson approximation. All are standard textbook exercises requiring only direct formula application with no problem-solving insight needed. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(\text{Type I error}) = P(1 \text{ or more}) = 0.0202\) | M1 | Identifying correct probability |
| A1 [2] | Correct answer (condone Poisson approx) | |
| (ii) \(P(\text{Type II error}) = P(0) \text{ under } H_1 = (1 - 0.0024)^2 = 0.972\) | B1 | Identifying correct probability |
| M1 | Attempt to find prob using 0.0024 | |
| A1 [3] | Correct final answer (condone Poisson approx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(2) = e^{-0.288}\left(\frac{0.288^2}{2}\right) = 0.0311\) | B1 | For 0.0024 × 120 in a Poisson expression |
| M1 | Poisson expression for P(2), any mean | |
| A1 [3] | Correct answer. SR Use of Binomial giving final answer of 0.0310 scores B1 |
(i) $P(\text{Type I error}) = P(1 \text{ or more}) = 0.0202$ | M1 | Identifying correct probability
| A1 [2] | Correct answer (condone Poisson approx)
(ii) $P(\text{Type II error}) = P(0) \text{ under } H_1 = (1 - 0.0024)^2 = 0.972$ | B1 | Identifying correct probability
| M1 | Attempt to find prob using 0.0024
| A1 [3] | Correct final answer (condone Poisson approx)
(iii) Poisson approximation $\lambda = 0.288$
$P(2) = e^{-0.288}\left(\frac{0.288^2}{2}\right) = 0.0311$ | B1 | For 0.0024 × 120 in a Poisson expression
| M1 | Poisson expression for P(2), any mean
| A1 [3] | Correct answer. SR Use of Binomial giving final answer of 0.0310 scores B1
---5 Every month Susan enters a particular lottery. The lottery company states that the probability, $p$, of winning a prize is 0.0017 each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her 12 lottery results in a one-year period. She accepts the null hypothesis $p = 0.0017$ if she has no wins in the year and accepts the alternative hypothesis $p > 0.0017$ if she wins a prize in at least one of the 12 months.\\
(i) Find the probability of the test resulting in a Type I error.\\
(ii) If in fact the probability of winning a prize each month is 0.0024 , find the probability of the test resulting in a Type II error.\\
(iii) Use a suitable approximation, with $p = 0.0024$, to find the probability that in a period of 10 years Susan wins a prize exactly twice.
\hfill \mbox{\textit{CAIE S2 2008 Q5 [8]}}