| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Binomial of Poisson approximations |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson approximation to binomial with routine conditions (n large, p small, np moderate), followed by basic Poisson probability calculations and a simple hypothesis test. All steps are textbook procedures requiring minimal problem-solving or insight, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial2.05a Hypothesis testing language: null, alternative, p-value, significance5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n > 50\) | B1 | Accept \(n\) large |
| \(np = 0.8\), which is \(< 5\) | B1 [2] | Accept \(p\) small |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lambda = 9.6\) | B1 | |
| \(e^{-9.6}\left(\dfrac{9.6^3}{3!} + \dfrac{9.6^4}{4!} + \dfrac{9.6^5}{5!}\right)\) | M1 | Any \(\lambda\). Accept end errors. |
| \(= 0.0800\) (3 sfs) | A1 [3] | Allow \(0.08\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): Pop mean for 10 days \(= 8\) \(\quad\) \(H_1\): Pop mean for 10 days \(< 8\) | B1 | or Pop mean for 1 day \(= 0.8\), Pop mean for 1 day \(< 0.8\). Allow \(\lambda\) or \(\mu\) but not just 'mean' |
| \(e^{-8}\!\left(1 + 8 + \dfrac{8^2}{2!}\right)\) | M1 | Any \(\lambda\). Accept end errors. NB P(2) only used scores M0M0. Accept CR method |
| \(= 0.0138\) or \(0.0137\) | A1 | \(CR = 0, 1, 2\) — all working must be shown |
| Compare \(0.02\); Evidence that mean number of absentees has decreased | M1 A1ft [5] | Valid comparison with \(0.02\) or CR. No contradictions. Reject \(H_0\) / accept \(H_1\) only if \(H_0\)/\(H_1\) correctly defined |
# Question 7:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $n > 50$ | B1 | Accept $n$ large |
| $np = 0.8$, which is $< 5$ | B1 [2] | Accept $p$ small |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lambda = 9.6$ | B1 | |
| $e^{-9.6}\left(\dfrac{9.6^3}{3!} + \dfrac{9.6^4}{4!} + \dfrac{9.6^5}{5!}\right)$ | M1 | Any $\lambda$. Accept end errors. |
| $= 0.0800$ (3 sfs) | A1 [3] | Allow $0.08$ |
## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: Pop mean for 10 days $= 8$ $\quad$ $H_1$: Pop mean for 10 days $< 8$ | B1 | or Pop mean for 1 day $= 0.8$, Pop mean for 1 day $< 0.8$. Allow $\lambda$ or $\mu$ but not just 'mean' |
| $e^{-8}\!\left(1 + 8 + \dfrac{8^2}{2!}\right)$ | M1 | Any $\lambda$. Accept end errors. NB P(2) only used scores M0M0. Accept CR method |
| $= 0.0138$ or $0.0137$ | A1 | $CR = 0, 1, 2$ — all working must be shown |
| Compare $0.02$; Evidence that mean number of absentees has decreased | M1 A1ft [5] | Valid comparison with $0.02$ or CR. No contradictions. Reject $H_0$ / accept $H_1$ only if $H_0$/$H_1$ correctly defined |
**Total: [10]**
**Total for paper: [50]**7 The number of workers, $X$, absent from a factory on a particular day has the distribution $\mathrm { B } ( 80,0.01 )$.\\
(i) Explain why it is appropriate to use a Poisson distribution as an approximating distribution for $X$.\\
(ii) Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 .
Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.\\
(iii) During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the $2 \%$ significance level.
\hfill \mbox{\textit{CAIE S2 2012 Q7 [10]}}