S2 June 2006 Q3
3. An estate agent sells properties at a mean rate of 7 per week.
(a) Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model. (3)
(b) Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties. (2)
(c) Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties. (6)
| Scheme | Marks |
|---|---|
| Let \(X\) represent the number of properties sold in a week \(\therefore X \sim \mathrm{Po}(7)\) | B1 |
| Sales occur independently/randomly, singly, at a constant rate | B1 B1 |
| (3) |
Notes
1st B1 must be in part a
B1 B1 context needed once
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 5) = \mathrm{P}(X \leqslant 5) - \mathrm{P}(X \leqslant 4) \quad\) or \(\dfrac{7^5\mathrm{e}^{-7}}{5!}\) | M1 |
| \(= 0.3007 - 0.1730\) \(= 0.1277\) | A1 |
| (2) |
Notes
A1 awrt 0.128
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 181) \approx \mathrm{P}(Y \geqslant 181.5)\) where \(Y \sim \mathrm{N}(168, 168)\) | B1 |
| \(= \mathrm{P}\left(z \geqslant \dfrac{181.5 - 168}{\sqrt{168}}\right)\) | M1 M1 |
| \(= \mathrm{P}(z \geqslant 1.04)\) | A1 |
| \(= 1 - 0.8508\) | M1 |
| \(= 0.1492\) | A1 |
| (6) | |
| (11 marks) |
Notes
B1 N(168, 168)
1st M1 \(\pm 0.5\)
2nd M1 stand with \(\mu\) and \(\sigma\)
1st A1 Give A1 for 1.04 or correct expression
3rd M1 attempt correct area \(1 - p\) where \(p \gt 0.5\)
2nd A1 awrt 0.149