| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Consecutive non-overlapping periods |
| Difficulty | Standard +0.3 This is a straightforward Poisson distribution question requiring standard techniques: solving P(X=0)=e^(-λ)=0.45 for λ, finding μ from sample data, then applying Poisson addition properties. Part (i) is routine algebra, part (ii) uses standard tables/calculation, and part (iii) requires recognizing that a fortnight uses parameter 2(λ+μ). All steps are textbook applications with no novel insight required, 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 probabilities5.06b Fit prescribed distribution: chi-squared test |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(e^a = 0.45\); \(\mu_G = 0.799 \approx 0.80\) AG | M1 | |
| A1 | 2 marks; 0.799 or 0.798 or better seen | |
| (ii) \(\mu_U \approx 1.8\) | B1 | |
| Total, \(T \sim Po(2.6)\); \(P(>3) = 0.264\) | M1 | May be implied by answer 0.264 |
| A1 | 3 marks; From table or otherwise | |
| (iii) \(e^{-2}2.6/6!\); \(e^{-3}5.2^3/41\) | B1 | Or 0.318 from table |
| B1 | ||
| Multiply two probabilities. Answers rounding to 0.0053 or 0.0054 | M1 | |
| A1 | 4 marks |
(i) $e^a = 0.45$; $\mu_G = 0.799 \approx 0.80$ AG | M1 |
| A1 | 2 marks; 0.799 or 0.798 or better seen
(ii) $\mu_U \approx 1.8$ | B1 |
Total, $T \sim Po(2.6)$; $P(>3) = 0.264$ | M1 | May be implied by answer 0.264
| A1 | 3 marks; From table or otherwise
(iii) $e^{-2}2.6/6!$; $e^{-3}5.2^3/41$ | B1 | Or 0.318 from table
| B1 |
Multiply two probabilities. Answers rounding to 0.0053 or 0.0054 | M1 |
| A1 | 4 marks |5 A music store sells both upright and grand pianos. Grand pianos are sold at random times and at a constant average weekly rate $\lambda$. The probability that in one week no grand pianos are sold is 0.45 .\\
(i) Show that $\lambda = 0.80$, correct to 2 decimal places.
Upright pianos are sold, independently, at random times and at a constant average weekly rate $\mu$. During a period of 100 weeks the store sold 180 upright pianos.\\
(ii) Calculate the probability that the total number of pianos sold in a randomly chosen week will exceed 3.\\
(iii) Calculate the probability that over a period of 3 weeks the store sells a total of 6 pianos during the first week and a total of 4 pianos during the next fortnight.
\hfill \mbox{\textit{OCR S3 2007 Q5 [9]}}