S2 June 2011 Q2
2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.
The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
| Scheme | Marks |
|---|---|
| Poisson | B1 |
| (1) |
Notes
B1 for Poisson or Po. Ignore their value for the mean.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 9\) (or \(\lambda = 36\)) \(\mathrm{H}_1 : \mu \gt 9\) (or \(\lambda \gt 36\)) | B1 B1 |
| \(X \sim \mathrm{Po}(9)\) and \(\mathrm{P}(X \geqslant 12) = 1 - \mathrm{P}(X \leqslant 11)\) or \(\mathrm{P}(X \leqslant 14) = 0.9585\) \(\mathrm{P}(X \geqslant 15) = 0.0415\) | M1 |
| \(= 1 - 0.8030 = \underline{0.197}\) CR \(X \geqslant 15\) | A1 |
| (0.197 > 0.05) so not significant/ accept H0/ Not in CR | M1d |
| he does not have evidence to switch on the speed restrictions (o.e) | A1ft |
| (6) |
Notes
1st B1 for \(\mathrm{H}_0 : \mu/\lambda = 9\) or \(\mu/\lambda = 36\)
2nd B1 for \(\mathrm{H}_1 : \mu/\lambda \gt 9\) or \(\mu/\lambda \gt 36\)
One tail
1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 11)\) or writing \(\mathrm{P}(X \leqslant 14) = 0.9585\) or \(\mathrm{P}(X \geqslant 15) = 0.0415\). May be implied by correct CR.or probability = 0.197
A1 for 0.197 or a correct CR. Allow \(X\) >14. NB \(\mathrm{P}(X \leqslant 11) = 0.8030\) on its own scores M1A1
2nd M1 dependent on the 1st M1 being awarded. For a correct statement based on the table below. Do not allow non-contextual conflicting statements eg “significant” and “accept H0”. Ignore comparisons.
2nd A1 for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.
| \(0.05 \lt p \lt 0.95\) | \(p \lt 0.05\) or \(p \gt 0.95\) | |
|---|---|---|
| 2nd M1 | not significant/ accept H0/ Not in CR | significant/ reject H0/ In CR |
| 2nd A1 | Insufficient evidence to switch on the speed restrictions | Sufficient evidence to switch on the speed restrictions |
Two tail
1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 11)\) or writing \(\mathrm{P}(X \leqslant 15) = 0.9780\) or \(\mathrm{P}(X \geqslant 16) = 0.022\). May be implied by correct CR. or probability = 0.197
A1 for 0.197 or CR \(X \geqslant 16\). Allow \(X\) >15. NB \(\mathrm{P}(X \leqslant 11) = 0.8030\) on its own scores M1A1
2nd M1 dependent on the 1st M1 being awarded . For a correct statement based on the table below. Do not allow non-contextual conflicting statements eg“significant” and “accept H0” . Ignore comparisons.
2nd A1 for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.
| \(0.025 \lt p \lt 0.975\) | \(p \lt 0.025\) or \(p \gt 0.975\) | |
|---|---|---|
| 2nd M1 | not significant/ accept H0/ Not in CR | significant/ reject H0/ In CR |
| 2nd A1 | Insufficient evidence to switch on the speed restrictions | Sufficient evidence to switch on the speed restrictions |
| Scheme | Marks |
|---|---|
| Let \(Y\) = the number of vehicles in 10 s then \(Y \sim \mathrm{Po}(6)\) | B1 |
| Tables: \(\mathrm{P}(Y \leqslant 10) = 0.9574\) so \(\mathrm{P}(Y \geqslant 11) = 0.0426\) | M1 |
| so needs 11 vehicles | A1 |
| (3) | |
| (10 marks) |
Notes
B1 for identifying Po(6) - may be implied by use of correct tables
M1 any one of the probs 0.9574 or 0.0426 or 0.9799 or 0.0201 may be implied by correct answer of 11
A1 cao do not accept \(X \geqslant 11\)
NB answer of 11 with no working gains all three marks.