S2 June 2008 Q5
5. Sue throws a fair coin 15 times and records the number of times it shows a head.
Find the probability that Sue records
Sue has a different coin which she believes is biased in favour of heads. She throws the coin 15 times and obtains 13 heads.
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(15, 0.5)\) | B1 B1 |
| (2) |
Notes
B1 for Binomial
B1 for 15 and 0.5 must be in part a
This need not be in the form written
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 8) = \mathrm{P}(X \leqslant 8) - \mathrm{P}(X \leqslant 7)\) or \(\left(\dfrac{15!}{8!\,7!}(p)^8(1 - p)^7\right)\) \(= 0.6964 - 0.5\) | M1 |
| \(= 0.1964\) awrt 0.196 | A1 |
| (2) |
Notes
M1 attempt to find \(\mathrm{P}(X = 8)\) any method. Any value of \(p\)
A1 awrt 0.196
Answer only full marks
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \geqslant 4) = 1 - \mathrm{P}(X \leqslant 3)\) \(= 1 - 0.0176\) | M1 |
| \(= 0.9824\) | A1 |
| (2) |
Notes
M1 for \(1 - \mathrm{P}(X \leqslant 3)\).
A1 awrt 0.982
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.5\) | B1 |
| \(\mathrm{H}_1 : p \gt 0.5\) | B1 |
| \(X \sim \mathrm{B}(15, 0.5)\) | |
| \(\mathrm{P}(X \geqslant 13) = 1 - \mathrm{P}(X \leqslant 12)\) \(\left[\mathrm{P}(X \geqslant 12) = 1 - 0.9824 = 0.0176\right]\) | M1 |
| \(= 1 - 0.9963\) \(\mathrm{P}(X \geqslant 13) = 1 - 0.9963 = 0.0037\) \(= 0.0037\) CR \(X \geqslant 13\) | A1 |
| \(0.0037 \lt 0.01\) \(13 \geqslant 13\) Reject \(\mathrm{H}_0\) or it is significant or a correct statement in context from their values | M1 |
| There is sufficient evidence at the 1% significance level that the coin is biased in favour of heads Or There is evidence that Sues belief is correct | A1 |
| (6) | |
| (12 marks) |
Notes
att \(\mathrm{P}(X \geqslant 13)\) awrt 0.0037/ CR \(X \geqslant 13\)
B1 for correct \(\mathrm{H}_0\). must use \(p\) or \(\pi\)
B1 for correct \(\mathrm{H}_1\) must be one tail must use \(p\) or \(\pi\)
M1 attempt to find \(\mathrm{P}(X \geqslant 13)\) correctly. E.g. \(1 - \mathrm{P}(X \leqslant 12)\)
A1 correct probability or CR
To get the next 2 marks the null hypothesis must state or imply that \((p) = 0.5\)
M1 for correct statement based on their probability or critical region or a correct contextualised statement that implies that. not just 13 is in the critical region.
A1 This depends on their M1 being awarded for rejecting \(\mathrm{H}_0\). Conclusion in context. Must use the words biased in favour of heads or biased against tails or sues belief is correct.
NB this is a B mark on EPEN.
They may also attempt to find \(\mathrm{P}(X \lt 13) = 0.9963\) and compare with 0.99