S4 January 2006 Q4
4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm{H}_0 : \lambda = 3\) and \(\mathrm{H}_1 : \lambda \gt 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x \gt 4\).
(a) Find the size of the test. (3)
The table gives the values of the power function of the test to two decimal places.
| \(\lambda\) | 4 | 5 | 6 | 7 |
| Power | \(r\) | 0.56 | \(s\) | 0.83 |
(b) Calculate the value of \(r\) and the value of \(s\). (2)
(c) Comment on the suitability of the test when \(\lambda = 4\). (1)
| Scheme | Marks |
|---|---|
| Size of test \(= \mathrm{P}(X \gt 4 \mid \lambda = 3)\) | M1 |
| \(= 1 - \mathrm{P}(X \leqslant 4 \mid \lambda = 3) = 1 - 0.8153\) | A1 |
| \(= 0.1847\) awrt 0.185 | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(r = 1 - 0.6288 = 0.3712 = 0.37\) (2 d.p.) | B1 |
| \(s = 1 - 0.2851 = 0.7149 = 0.71\) (2 d.p.) | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| When \(\lambda = 4\), power \(= 0.37 \lt 0.5\) Probability of coming to correct conclusion is less than probability of coming to wrong conclusion. Not suitable. | B1 |
| (1) | |
| (6 marks) |
Notes
(corrected from the printed mark scheme: the scheme prints “power = 0.27”; part (b) gives \(r = 0.37\))