S3 June 2005 Q5
5. The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table.
| Number of restarts | Frequency |
|---|---|
| 0 | 99 |
| 1 | 65 |
| 2 | 22 |
| 3 | 12 |
| 4 | 2 |
Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a 5% level of significance and state your hypothesis clearly. (12)
| Scheme | Marks | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\mathrm{H}_0\): Poisson distribution is a suitable model \(\mathrm{H}_1\): Poisson distribution is not a suitable model both | B1 | ||||||||||||||||||||||||||||
| \(\hat{\lambda} = \dfrac{(0 \times 99) + (1 \times 65) + \cdots + (4 \times 2)}{200} = \dfrac{153}{200} = \underline{0.765}\) | M1 A1 | ||||||||||||||||||||||||||||
| Using \(\mathrm{P}(X = x) = \dfrac{0.765^x \mathrm{e}^{-0.765}}{x!}\) where \(X\) represents the number of restarts gives \(200 \times \mathrm{P}(X = x)\) | M1 | ||||||||||||||||||||||||||||
| A1, A1 (−1 e.e.) A1 | ||||||||||||||||||||||||||||
| \(\nu = 4 - 1 - 1 = 2\); CR: \(\chi^2_2 \gt 5.991\) from Poisson \(\nu = 4 - 1 = 3\); CR: \(\chi^2 \gt 7.815\) from Poisson(0.765) | B1; B1ft | ||||||||||||||||||||||||||||
| \(\displaystyle\sum \frac{(O - E)^2}{E} = 5.47368\ldots\) Use of \(\sum (O - E)^2/E\) | M1 | ||||||||||||||||||||||||||||
| 5.40 – 5.50 | A1 | ||||||||||||||||||||||||||||
| 5.47 is not in the critical region. Number of computer failures per day can be modelled by a Poisson distribution | A1ft | ||||||||||||||||||||||||||||
| (12) | |||||||||||||||||||||||||||||
| (12 marks) |