| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Poisson approximation to binomial |
| Difficulty | Moderate -0.8 This is a straightforward application of Poisson approximation to binomial with standard probability calculations. The setup is clear (n=80, p=0.02 gives λ=1.6), and both parts require only direct use of Poisson tables or formulas with no conceptual challenges or multi-step reasoning—easier than average A-level questions. |
| 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 probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - \text{P}(\leq 1) = 1 - 0.5249 = 0.4751\) | Marks: M1 M1 M1 A1 | Guidance: \(\text{B}(80, 0.02)\) seen or implied, e.g. \(\text{N}(1.6, 1.568)\). Po(\(np\)) used. \(1 - \text{P}(\leq 1)\) used. Answer, a.r.t. 0.475. [SR: Exact: M1 M0 M0, 0.477 A1] |
| Answer | Marks | Guidance |
|---|---|---|
| Therefore least value is 6 | Marks: M1 A1 A1 | Guidance: Evidence for correct method, e.g. answer 6. At least one of these probabilities seen. Answer 6 only. [SR N(1.6,1.568): 2.326 = (r – 1.6)/√1.568 M1; r = 5 or (with cc) 6 A1; Exact: M1 A0 A1] |
**(i)** **Answer:** $\text{B}(80, 0.02)$ approx $\text{Po}(1.6)$
$1 - \text{P}(\leq 1) = 1 - 0.5249 = 0.4751$ | **Marks:** M1 M1 M1 A1 | **Guidance:** $\text{B}(80, 0.02)$ seen or implied, e.g. $\text{N}(1.6, 1.568)$. Po($np$) used. $1 - \text{P}(\leq 1)$ used. Answer, a.r.t. 0.475. [SR: Exact: M1 M0 M0, 0.477 A1]
**(ii)** **Answer:** $\text{P}(\leq 4) = 0.9763, \text{P}(\leq 5) = 0.0237$
$\text{P}(\leq 5) = 0.9940, \text{P}(\leq 6) = 0.0060$
Therefore least value is 6 | **Marks:** M1 A1 A1 | **Guidance:** Evidence for correct method, e.g. answer 6. At least one of these probabilities seen. Answer 6 only. [SR N(1.6,1.568): 2.326 = (r – 1.6)/√1.568 M1; r = 5 or (with cc) 6 A1; Exact: M1 A0 A1]4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by $R$.\\
(i) Use an appropriate approximation to find $\mathrm { P } ( R \geqslant 2 )$.\\
(ii) Find the smallest value of $r$ for which $\mathrm { P } ( R \geqslant r ) < 0.01$.
\hfill \mbox{\textit{OCR S2 Q4 [7]}}