| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a standard hypothesis testing question covering routine Poisson test procedures (finding critical region, conducting test, Type I error) plus a normal approximation to sum of Poissons. All techniques are textbook applications with no novel insight required, though the multi-part structure and normal approximation push it slightly above average difficulty. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): Pop mean (or \(\lambda\) or \(\mu\)) is 5.3; \(H_1\): Pop mean (or \(\lambda\) or \(\mu\)) is less than 5.3 | B1 | Both |
| \(P(X \leq 1) = e^{-5.3}(1 + 5.3)\) and \(P(X \leq 2) = e^{-5.3}(1 + 5.3 + \frac{5.3^2}{2})\) / \(P(X=2)\) | M1 | Both attempted |
| \(P(X \leq 1) = 0.0314\) or \(0.0315\) & \(P(X \leq 2) = 0.102\)/ \(P(X=2)=0.7071\) | A1 | Both correct |
| CR is 0 or 1 cases | A1 | Dep. M1 and \(P(X \leq 1) < 0.05 < P(X \leq 2)\) |
| No evidence mean has decreased | B1f [5] | ft their CR |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Concluding mean has decreased when it hasn't | B1 | In context |
| '\(0.0314\) or \(0.0315\)' | B1ft[2] | ft their \(P(X \leq 1)\), dep. \(< 0.05\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((Po(18.4))\) \(N(18.4, 18.4)\) | B1, B1ft | Stated or implied; B1 for \(N(18.4, ...)\); B1ft for var. \(= 18.4\) |
| \(\frac{20.5 - 18.4}{\sqrt{18.4}} \quad (= 0.490)\) | M1 | For standardising with or without cc. Allow without \(\sqrt{\phantom{0}}\) |
| \(1 - \Phi('0.490')\) | M1 | Use of tables and attempt to find area consistent with their working |
| \(= 0.312\) (3 s.f.) | A1 [5] |
## Question 6:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: Pop mean (or $\lambda$ or $\mu$) is 5.3; $H_1$: Pop mean (or $\lambda$ or $\mu$) is less than 5.3 | B1 | Both |
| $P(X \leq 1) = e^{-5.3}(1 + 5.3)$ and $P(X \leq 2) = e^{-5.3}(1 + 5.3 + \frac{5.3^2}{2})$ / $P(X=2)$ | M1 | Both attempted |
| $P(X \leq 1) = 0.0314$ or $0.0315$ & $P(X \leq 2) = 0.102$/ $P(X=2)=0.7071$ | A1 | Both correct |
| CR is 0 or 1 cases | A1 | Dep. M1 and $P(X \leq 1) < 0.05 < P(X \leq 2)$ |
| No evidence mean has decreased | B1f [5] | ft their CR |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Concluding mean has decreased when it hasn't | B1 | In context |
| '$0.0314$ or $0.0315$' | B1ft[2] | ft their $P(X \leq 1)$, dep. $< 0.05$ |
### Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(Po(18.4))$ $N(18.4, 18.4)$ | B1, B1ft | Stated or implied; B1 for $N(18.4, ...)$; B1ft for var. $= 18.4$ |
| $\frac{20.5 - 18.4}{\sqrt{18.4}} \quad (= 0.490)$ | M1 | For standardising with or without cc. Allow without $\sqrt{\phantom{0}}$ |
| $1 - \Phi('0.490')$ | M1 | Use of tables and attempt to find area consistent with their working |
| $= 0.312$ (3 s.f.) | A1 [5] | |
**[Total: 12]**6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the $5 \%$ significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.\\
(i) Find the critical region for the test and, given that the number of cases is 2 , carry out the test.\\
(ii) Explain the meaning of a Type I error in this context and state the probability of a Type I error.\\
(iii) At another clinic the mean number of cases of asthma per month has the independent distribution $\mathrm { Po } ( 13.1 )$. Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.
\hfill \mbox{\textit{CAIE S2 2013 Q6 [12]}}