S2 June 2006 Q5
5. A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6% of the production will be green.
A random sample of 10 mugs was taken from the production line.
(a) Define a suitable distribution to model the number of green mugs in this sample. (1)
(b) Find the probability that there were exactly 3 green mugs in the sample. (3)
A random sample of 125 mugs was taken.
(c) Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
(i) a Poisson approximation, (3)
(ii) a Normal approximation. (6)
| Scheme | Marks |
|---|---|
| Binomial | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Let \(X\) represent the number of green mugs in a sample \(X \sim \mathrm{B}(10, 0.06)\) | B1 |
| \(\mathrm{P}(X = 3) = {}^{10}\mathrm{C}_3(0.06)^3(0.94)^7\) | M1 |
| \(= 0.016808\ldots\) | A1 |
| (3) |
Notes
B1 may be implied or seen in part a
M1 \({}^{10}\mathrm{C}_3(p)^3(1 - p)^7\)
A1 awrt 0.0168
| Scheme | Marks |
|---|---|
| Let \(X\) represent number of green mugs in a sample of size 125 (i) \(X \sim \mathrm{Po}(125 \times 0.06 = 7.5)\) | B1 |
| \(\mathrm{P}(10 \leqslant X \leqslant 13) = \mathrm{P}(X \leqslant 13) - \mathrm{P}(X \leqslant 9)\) \(= 0.9784 - 0.7764\) | M1 |
| \(= 0.2020\) | A1 |
| (3) | |
| (ii) \(\mathrm{P}(10 \leqslant X \leqslant 13) \approx \mathrm{P}(9.5 \leqslant Y \leqslant 13.5)\) where \(Y \sim \mathrm{N}(7.5, 7.05)\) | B1 B1 |
| \(= \mathrm{P}\left(\dfrac{9.5 - 7.5}{\sqrt{7.05}} \leqslant z \leqslant \dfrac{13.5 - 7.5}{\sqrt{7.05}}\right)\) | M1 M1 |
| \(= \mathrm{P}(0.75 \leqslant z \leqslant 2.26)\) | A1 |
| \(= 0.2147\) | A1 |
| (6) | |
| (13 marks) |
Notes
(i) B1 may be implied
(i) A1 awrt 0.202
(ii) 1st B1 7.05
(ii) 2nd B1 9.5, 13.5
(ii) 1st M1 \(\pm 0.5\)
(ii) 2nd M1 stand.
(ii) 1st A1 both values or both correct expressions. awrt 0.75 and 2.26
(ii) 2nd A1 awrt 0.214 or 0.215