| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 3 |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Simple probability using normal approximation |
| Difficulty | Moderate -0.3 This is a straightforward application of the normal approximation to the Poisson distribution with clear parameters (λ=25). Part (i) requires a standard probability calculation with continuity correction, and part (ii) involves finding an inverse normal value. Both are routine techniques for this topic with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem |
Use of N(25, 25) (in either part)
(i) $z = \frac{15.5 - 25}{5} = -1.9$ **M1, A1**
$P(\leqslant 15) = 0.0287$ **A1** [4 marks total... but this is part (i)]
(ii) $\Phi(z) = 0.95 \Rightarrow z = 1.645$ (Allow [1.64, 1.65]) **B1**
$x = 0.5 + 25 + 5 \times 1.645 = 33.725$ (Allow for $\pm 0.5$) **B1**
$\Rightarrow$ 34 narrowboats required. (CAO) **B1** [3]1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find\\
(i) the probability that 15 or fewer narrowboats are hired out during a certain week,\\
(ii) the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q1 [3]}}