S2 January 2012 Q7
7.
The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
| Scheme | Marks |
|---|---|
| (i) The range of values/region/area/set of values of the test statistic that would lead you to reject H0 | B1 |
| (ii) The probability of incorrectly rejecting \(\mathrm{H}_0\) or Probability of rejecting \(\mathrm{H}_0\) when \(\mathrm{H}_0\) is true | B1 |
| (2) |
Notes
(i) Allow accept H1 instead of reject H0. It must be clear which hypothesis gets rejected/accepted.
(ii) Allow equivalent wording.
| Scheme | Marks |
|---|---|
| (i) \(X \sim \mathrm{Po}(8)\) | M1 |
| \(\mathrm{P}(X \leqslant 4) = 0.0996\) \(\mathrm{P}(X \leqslant 3) = 0.0424\) Critical region [0,3] | A1 |
| (ii) awrt 0.0424 | B1 |
| (3) |
Notes
M1 Writing or using Po(8). May be implied by correct critical region.
A1 allow \(0 \leqslant X \leqslant 3\) or CR \(\leqslant 3\) or \(X \leqslant 3\). Any letter may be used but not \(\mathrm{P}(X \leqslant 3)\). This must be on its own.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 8\) (or \(\mu = 8\)) \(\mathrm{H}_1 : \lambda \gt 8\) (or \(\mu \gt 8\)) | B1 |
| \(\mathrm{P}(X \geqslant 13) = 1 - \mathrm{P}(X \leqslant 12)\) or \(\mathrm{P}(X \leqslant 13) = 0.9658\) or \(\mathrm{P}(X \geqslant 14) = 0.0342\) | M1 |
| \(= 1 - 0.9362\) \(= 0.0638\) CR \(X \geqslant 14\) | A1 |
| so insufficient evidence to reject \(\mathrm{H}_0\) /not significant/ not in critical region | M1 dep |
| There in insufficient evidence of an increase/change in the rate/number of sales per month or the estate agents claim is incorrect | A1 |
| (5) | |
| (10 marks) |
Notes
B1 both hypotheses correct. Must use \(\lambda\) or \(\mu\).
One tail
1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 12)\) or writing \(\mathrm{P}(X \leqslant 13) = 0.9658\) or \(\mathrm{P}(X \geqslant 14) = 0.0342\). May be implied by correct CR.or probability = 0.0638
A1 for 0.0638 or \(X \geqslant 14\). Allow \(X\) >13. NB \(\mathrm{P}(X \leqslant 12) = 0.9362\) 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 “not significant” and “reject 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 of an increase/change in the rate/number of sales per month | Sufficient evidence of an increase/change in the rate/number of sales per month |
Two tail
1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 12)\) or writing \(\mathrm{P}(X \leqslant 14) = 0.9827\) or \(\mathrm{P}(X \geqslant 15) = 0.0173\). May be implied by correct CR.or probability = 0.0638
A1 for 0.0638 or \(X \geqslant 15\). Allow \(X\) >14. NB \(\mathrm{P}(X \leqslant 12) = 0.9362\) 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 “not significant” and “reject 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 of an increase/change in the rate/number of sales per month | Sufficient evidence of an increase/change in the rate/number of sales per month |