S2 January 2009 Q6
6. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
(a)
(i) Test, at the 10% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
(ii) State the minimum number of visits required to obtain a significant result. (7)
(b) State an assumption that has been made about the visits to the server. (1)
In a random two minute period on a Saturday the web server is visited 20 times.
(c) Using a suitable approximation, test at the 10% level of significance, whether or not the rate of visits is greater on a Saturday. (6)
| Scheme | Marks |
|---|---|
| (i) \(\mathrm{H}_0 : \lambda = 7 \qquad \mathrm{H}_1 : \lambda \gt 7\) | B1 |
| \(X\) = number of visits. \(X \sim \mathrm{Po}(7)\) | B1 |
| \(\mathrm{P}(X \geqslant 10) = 1 - \mathrm{P}(X \leqslant 9)\) \(1 - \mathrm{P}(X \leqslant 10) = 0.0985\) | M1 |
| \(= 0.1695\) \(1 - \mathrm{P}(X \leqslant 9) = 0.1695\) CR \(X \geqslant 11\) | A1 |
| \(0.1695 \gt 0.10\), CR \(X \geqslant 11\) Not significant or it is not in the critical region or do not reject \(\mathrm{H}_0\) | M1 |
| The rate of visits on a Saturday is not greater/ is unchanged | A1 no ft |
| (ii) \(X = 11\) | B1 |
| (7) |
| Scheme | Marks |
|---|---|
| (The visits occur) randomly/ independently or singly or constant rate | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \([\mathrm{H}_0 : \lambda = 7 \qquad \mathrm{H}_1 : \lambda \gt 7 \quad (\text{or } \mathrm{H}_0 : \lambda = 14 \qquad \mathrm{H}_1 : \lambda \gt 14)]\) | |
| \(X \sim \mathrm{N};(14, 14)\) | B1;B1 |
| \(\mathrm{P}(X \geqslant 20) = \mathrm{P}\left(z \geqslant \dfrac{19.5 - 14}{\sqrt{14}}\right)\) +/- 0.5, stand | M1 M1 |
| \(= \mathrm{P}(z \geqslant 1.47)\) \(= 0.0708\) or \(z = 1.2816\) | A1dep both M |
| \(0.0708 \lt 0.10\) therefore significant. The rate of visits is greater on a Saturday | A1dep 2nd M |
| (6) | |
| (14 marks) |