S2 January 2008 Q7
7.
During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
In the school holidays, 1 call occurs in a 10 minute interval.
| Scheme | Marks |
|---|---|
| (i) A hypothesis test is a mathematical procedure to examine a value of a population parameter proposed by the null hypothesis compared with an alternative hypothesis. | B1 |
| (ii) The critical region is the range of values or a test statistic or region where the test is significant | B1g |
| that would lead to the rejection of \(\mathrm{H}_0\). | B1h |
| (3) |
Notes
(i) B1 Method for deciding between 2 hypothesis.
(ii) B1 range of values. This may be implied by other words. Not region on its own
B1 which lead you to reject \(\mathrm{H}_0\)
Give the first B1 if only one mark awarded.
| Scheme | Marks |
|---|---|
| Let X represent the number of incoming calls : \(X \sim \mathrm{Po}(9)\) | B1 |
| From table \(\mathrm{P}(X \geqslant 16) = 0.0220\) | M1 A1 |
| \(\mathrm{P}(x \leqslant 3) = 0.0212\) | A1 |
| Critical region (\(x \leqslant 3\) or \(x \geqslant 16\)) | B1 |
| (5) |
Notes
B1 using \(\mathrm{P_o}(9)\)
M1 attempting to find \(\mathrm{P}(X \geqslant 16)\) or \(\mathrm{P}(x \leqslant 3)\)
A1 0.0220 or \(\mathrm{P}(X \geqslant 16)\)
A1 0.0212 or \(\mathrm{P}(x \leqslant 3)\)
These 3 marks may be gained by seeing the numbers in part c
B1 correct critical region
A completely correct critical region will get all 5 marks.
Half of the correct critical region eg \(x \leqslant 3\) or \(x \geqslant 17\) say would get B1 M1 A0 A1 B0 if the M1 A1 A1 not already awarded.
| Scheme | Marks |
|---|---|
| Significance level \(= 0.0220 + 0.0212\) \(= 0.0432\) or 4.32% | B1 |
| (1) |
Notes
B1 cao awrt 0.0432
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 0.45;\quad \mathrm{H}_1 : \lambda \lt 0.45\) (accept : \(\mathrm{H}_0 : \lambda = 4.5;\quad \mathrm{H}_1 : \lambda \lt 4.5\)) | B1 |
| Using \(X \sim \mathrm{Po}(4.5)\) | M1 |
| \(\mathrm{P}(X \leqslant 1) = 0.0611\) CR \(X \leqslant 0\) awrt 0.0611 | A1 |
| \(0.0611 \gt 0.05.\) \(1 \geqslant 0\) or 1 not in the critical region There is evidence to Accept \(\mathrm{H}_0\) or it is not significant | M1 |
| There is no evidence that there are less calls during school holidays. | B1cao |
| (5) | |
| (14 marks) |
Notes
B1 may use \(\lambda\) or \(\mu\). Needs both \(\mathrm{H}_0\) and \(\mathrm{H}_1\)
M1 using \(\mathrm{P_o}(4.5)\)
A1 correct probability or CR only
M1 correct statement based on their probability, \(\mathrm{H}_1\) and 0.05
or a correct contextualised statement that implies that.
B1 this is not a follow through. Conclusion in context. Must see the word calls in conclusion
If they get the correct CR with no evidence of using \(\mathrm{P_o}(4.5)\) they will get M0 A0
SC If they get the critical region \(X \leqslant 1\) they score M1 for rejecting \(\mathrm{H}_0\) and B1 for concluding the rate of calls in the holiday is lower.