| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - large lambda direct |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with rate scaling (parts i-ii) and normal approximation to Poisson (part iii). While it requires understanding of rate adjustment and continuity correction, these are standard S2 techniques with no novel problem-solving required. Slightly above average due to the multi-part nature and need to apply normal approximation correctly. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | ||
| \(e^{-3.34} \times \frac{3.84^4}{4!}\) | M1 | Poisson \(P(X = 4)\), any \(\lambda\) |
| \(= 0.195\) (3 sf) | A1 | [2] |
| Part (ii) | ||
| 1.44 | B1 | Seen |
| \(1 - e^{-1.44}\left(1 + 1.44 + \frac{1.44^2}{2}\right)\) | M1 | Any \(\lambda\), allow one end error, need "1 −..." |
| \(= 0.176\) | A1 | [3] |
| Part (iii) | ||
| \(X \sim N(41, 41)\) | B1 | Seen or implied |
| \(\frac{40.5 - 41}{\sqrt{41}}(= -0.078)\) \(\frac{59.5 - 41}{\sqrt{41}}(= 2.889)\) | M1M1 | M1M0 if no cc or incorrect cc OR no \(\sqrt{}\) in both |
| \(\Phi('2.889') - \Phi('-0.078')\) | M1 | Use of tables and correct area consistent with their working. |
| \(= \Phi(2.889') - (1 - \Phi('0.078'))\) | ||
| \(= 0.9981 - (1 - 0.5311)\) | ||
| \(= 0.529\) (3sf) | A1 | [5] cwo |
| Total | [10] |
| **Part (i)** |
|---|
| $e^{-3.34} \times \frac{3.84^4}{4!}$ | M1 | Poisson $P(X = 4)$, any $\lambda$ |
| $= 0.195$ (3 sf) | A1 | [2] |
| **Part (ii)** |
|---|
| 1.44 | B1 | Seen |
| $1 - e^{-1.44}\left(1 + 1.44 + \frac{1.44^2}{2}\right)$ | M1 | Any $\lambda$, allow one end error, need "1 −..." |
| $= 0.176$ | A1 | [3] |
| **Part (iii)** |
|---|
| $X \sim N(41, 41)$ | B1 | Seen or implied |
| $\frac{40.5 - 41}{\sqrt{41}}(= -0.078)$ $\frac{59.5 - 41}{\sqrt{41}}(= 2.889)$ | M1M1 | M1M0 if no cc or incorrect cc OR no $\sqrt{}$ in both |
| $\Phi('2.889') - \Phi('-0.078')$ | M1 | Use of tables and correct area consistent with their working. |
| $= \Phi(2.889') - (1 - \Phi('0.078'))$ | | |
| $= 0.9981 - (1 - 0.5311)$ | | |
| $= 0.529$ (3sf) | A1 | [5] cwo |
| **Total** | [10] |6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of\\
(i) exactly 4 calls in an 8 -minute period,\\
(ii) at least 3 calls in a 3-minute period.
The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.\\
(iii) Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.
\hfill \mbox{\textit{CAIE S2 2014 Q6 [10]}}