S2 June 2006 Q4
4. Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
(a) Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week. (4)
Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
(b) Test, at the 5% level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly. (7)
| Scheme | Marks |
|---|---|
| Let \(X\) represent the number of breakdowns in a week. \(X \sim \mathrm{Po}(1.25)\) | B1 |
| \(\mathrm{P}(X \lt 3) = \mathrm{P}(0) + \mathrm{P}(1) + \mathrm{P}(2) \quad\) or \(\mathrm{P}(X \leqslant 2)\) | M1 |
| \(= \mathrm{e}^{-1.25}\left(1 + 1.25 + \dfrac{(1.25)^2}{2!}\right)\) | A1 |
| \(= 0.868467\ldots\) | A1 |
| (4) |
Notes
B1 implied
2nd A1 awrt 0.868 or 0.8685
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0: \lambda = 1.25;\ \ \mathrm{H}_1: \lambda \neq 1.25 \quad\) (or \(\mathrm{H}_0: \lambda = 5;\ \ \mathrm{H}_1: \lambda \neq 5\)) | B1 B1 |
| Let \(Y\) represent the number of breakdowns in 4 weeks Under \(\mathrm{H}_0\), \(Y \sim \mathrm{Po}(5)\) | B1 |
| \(\mathrm{P}(Y \geqslant 11) = 1 - \mathrm{P}(Y \leqslant 10) \quad\) or \(\mathrm{P}(X \geqslant 11) = 0.0137\) \(\mathrm{P}(X \geqslant 10) = 0.0318\) | M1 |
| \(= 0.0137 \qquad \text{CR } X \geqslant 11\) | A1 |
| \(0.0137 \lt 0.025\), \(0.0274 \lt 0.05\), \(0.9863 \gt 0.975\), \(0.9726 \gt 0.95\) or \(11 \geqslant 11\) | M1 |
| Evidence that the rate of breakdowns has changed /decreased | B1ft |
| (7) | |
| (11 marks) |
Notes
B1 B1 \(\lambda\) or \(\mu\)
3rd B1 may be implied
1st M1 one needed for M
2nd M1 any, allow %; ft from \(\mathrm{H}_1\)
B1ft context; from their p