S2 January 2011 Q4
4. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard’s claim at the 5% level of significance. State your hypotheses clearly. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0\): \(\lambda = 8\) or \(\mu = 2\) \(\mathrm{H}_1\): \(\lambda \lt 8\) or \(\mu \lt 2\) | B1 B1 |
| Under \(\mathrm{H}_0\), \(X \sim \mathrm{Po}(8)\) | M1 |
| \(\mathrm{P}(X \leqslant 3) = 0.0424\) CR \(X \leqslant 3\) | A1 |
| \(0.0424 \lt 0.05\), Reject \(\mathrm{H}_0\). Richard’s claim is supported. | M1A1ft |
| (6 marks) |
Notes
B1 for H0 correct. Must use \(\lambda\) or \(\mu\) and 8 or 2
B1 for H1 correct. Must use \(\lambda\) or \(\mu\) and 8 or 2
M1 for writing or using Po(8) – may be implied by correct CR
A1 awrt 0.0424 or CR \(X \leqslant 3\)
M1 need \(p\)<0.5 and:
correct statement using their Probability and 0.05 if one tail test or
correct statement using their Probability and 0.025 if two tail test (condone a comparison with 0.05 instead of 0.025 for a two tail test).
Do not allow non-contextual conflicting statements eg “significant” and “accept H0”
A1ft correct contextual statement followed through from “their prob”.
Either a comment on whether Richard’s claim was correct or on whether the service has improved.
NB if a correct contextual statement only is given for their probability then award M1 A1
\(p\)>0.5
They may compare with 0.95 (one tail method) or 0.975 (two tail method)
Probability is 0.9576