| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.5 This is a straightforward application of standard distribution approximations. Part (a) requires recognizing when Poisson approximates Binomial (large n, small p) and calculating a basic probability. Part (b) is a routine one-tailed binomial hypothesis test. Both parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Po}(0.025)\) | B1 | For Poisson and correct parameter. |
| \(n = 2500 > 50\), \(np = 0.025 < 5\) | B1 | Must show 2500 and 0.025. Accept \(p = \frac{1}{100000} < 0.1\) in place of \(np = 0.025 < 5\). |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - e^{-0.025}\) | M1 | Allow any \(\lambda\). FT *their* (a)(i) if normal; must have continuity correction. |
| \(0.0247\) (3sf) | A1 | Must be from Poisson. Unsupported correct answer scores B1 instead of M1 A1. |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p = 0.3\); \(H_1: p < 0.3\) | B1 | |
| \(0.7^{28} + 28 \times 0.7^{27} \times 0.3 + {}^{28}C_2 \times 0.7^{26} \times 0.3^2 + {}^{28}C_3 \times 0.7^{25} \times 0.3^3 + {}^{28}C_4 \times 0.7^{24} \times 0.3^4\) | M1 | Use of \(B(28, 0.3)\). Addition of terms must be intended. Allow one term wrong or omitted or extra. |
| \(0.0474\) | A1 | Unsupported correct answer scores B1 instead of M1 A1. |
| \(0.0474 > 0.02\) [Not reject \(H_0\)] | M1 | Valid comparison. |
| No evidence that suspicion is true. | A1ft | Not definite e.g. not 'Suspicion is not true', in context, no contradictions. SC use of \(N(8.4, 5.88)\) leading to \(0.054 > 0.2\) OE can score B1 only for comparison and correct conclusion. Correct hypotheses with \(p\) will also score B1. |
| 5 |
## Question 5(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}(0.025)$ | B1 | For Poisson and correct parameter. |
| $n = 2500 > 50$, $np = 0.025 < 5$ | B1 | Must show 2500 and 0.025. Accept $p = \frac{1}{100000} < 0.1$ in place of $np = 0.025 < 5$. |
| | **2** | |
---
## Question 5(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - e^{-0.025}$ | M1 | Allow any $\lambda$. FT *their* **(a)(i)** if normal; must have continuity correction. |
| $0.0247$ (3sf) | A1 | Must be from Poisson. Unsupported correct answer scores **B1** instead of **M1 A1**. |
| | **2** | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = 0.3$; $H_1: p < 0.3$ | B1 | |
| $0.7^{28} + 28 \times 0.7^{27} \times 0.3 + {}^{28}C_2 \times 0.7^{26} \times 0.3^2 + {}^{28}C_3 \times 0.7^{25} \times 0.3^3 + {}^{28}C_4 \times 0.7^{24} \times 0.3^4$ | M1 | Use of $B(28, 0.3)$. Addition of terms must be intended. Allow one term wrong or omitted or extra. |
| $0.0474$ | A1 | Unsupported correct answer scores **B1** instead of **M1 A1**. |
| $0.0474 > 0.02$ [Not reject $H_0$] | M1 | Valid comparison. |
| No evidence that suspicion is true. | A1ft | Not definite e.g. not 'Suspicion is not true', in context, no contradictions. SC use of $N(8.4, 5.88)$ leading to $0.054 > 0.2$ OE can score **B1** only for comparison and correct conclusion. Correct hypotheses with $p$ will also score B1. |
| | **5** | |
---5
\begin{enumerate}[label=(\alph*)]
\item The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by $X$.
\begin{enumerate}[label=(\roman*)]
\item State, with a justification, a suitable approximating distribution for $X$, giving the values of any parameters.
\item Use the approximating distribution to calculate $\mathrm { P } ( X > 0 )$.
\end{enumerate}\item The percentage of people having a different medical condition is thought to be $30 \%$. A researcher suspects that the true percentage is less than $30 \%$. In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition.
Use a binomial distribution to test the researcher's suspicion at the $2 \%$ significance level.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q5 [9]}}