S2 June 2012 Q2
2. A test statistic has a distribution B(25, \(p\)).
Given that
\[\mathrm{H}_0 : p = 0.5 \qquad \mathrm{H}_1 : p \ne 0.5\]| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(25, 0.5)\) may be implied by calculations in part a or b | M1 |
| \(\mathrm{P}(X \leqslant 7) = 0.0216\) | |
| \(\mathrm{P}(X \geqslant 18) = 0.0216\) | |
| CR \(X \leqslant 7\); \(\cup\) \(X \geqslant 18\) | A1,A1 |
| (3) |
Notes
M1 – Using B(25,0.5) – may be implied by a correct critical region or by calculations in part a or b
Note Just seeing either \(\mathrm{P}(X \leqslant 7)\) or \(\mathrm{P}(X \geqslant 18)\) scores M1 A0 A0.
You may need to check their probabilities in the tables for values other than 7 or 18.
1st A1 – also allow \(X \lt 8\) or [0,7] or \(0 \leqslant X \leqslant 7\) or \(0 \leqslant X \lt 8\) oe e.g. [0, 8) or a full list
DO NOT allow CRs given as \(\mathrm{P}(X \leqslant 7)\) or 7 – 0 for the A mark.
2nd A1 – also allow \(X \gt 17\) or [18,25] or \(18 \leqslant X \leqslant 25\) or \(17 \lt X \leqslant 25\) oe e.g. (17, 25] or a full list
DO NOT allow CRs given as \(\mathrm{P}(X \geqslant 18)\) or 18 - 25 for the A mark.
SC \(7 \geqslant X \geqslant 18\) gains M1 A1 A0.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{rejecting } \mathrm{H}_0) = 0.0216 + 0.0216\) | M1 |
| \(= 0.0432\) awrt 0.0432/0.0433 | A1 |
| (2) | |
| (5 marks) |
Notes
M1 – adding their two critical regions’ probabilities together or may be awarded for awrt 0.0432
If they add their critical regions’ probabilities and then go on and get a different probability as their answer then it is M0A0
e.g. \(0.0216 + 0.0216 = 0.0432\) then \(0.05 - 0.0432 = 0.0068\) gets M0 A0
e.g. \(0.0216 + 0.0216 = 0.0432 \lt 0.05\) reject \(\mathrm{H}_0\) gets M1 A1
e.g. \(0.0216 + 0.0216 = 0.0432\) so probability of rejecting \(\mathrm{H}_0\) is \(1 - 0.0432 = 0.9568\) gets M0 A0