S2 June 2016 Q2
2. In a region of the UK, 5% of people have red hair. In a random sample of size \(n\), taken from this region, the expected number of people with red hair is 3
A random sample of 20 people is taken from this region.
Find the probability that
Patrick claims that Reddman people have a probability greater than 5% of having red hair. In a random sample of 50 Reddman people, 4 of them have red hair.
| Scheme | Marks |
|---|---|
| \(0.05n = 3\) | M1 |
| \(n = 60\) | A1 |
| (2) |
Notes
M1: using \(0.05n\)
A1: cao
NB: for 60 with no incorrect working award M1A1
| Scheme | Marks |
|---|---|
| \(R \sim \mathrm{B}(20, 0.05)\) | B1 |
| (i) \(\mathrm{P}(R = 4) = {}^{20}C_4(0.05)^4(0.95)^{16}\) OR \(\mathrm{P}(R = 4) = \mathrm{P}(R \leqslant 4) - \mathrm{P}(R \leqslant 3)\) \(= 0.9974 - 0.9841\) | M1 |
| \(= 0.0133\) | A1 |
| (ii) \(\mathrm{P}(R \geqslant 4) = 1 - \mathrm{P}(R \leqslant 3)\) \(= 1 - 0.9841\) | M1 |
| \(= 0.0159\) | A1 |
| (5) |
Notes
B1: using or writing B(20, 0.05) in (i) or (ii)
(i) M1 writing or using \(\mathrm{P}(R \leqslant 4) - \mathrm{P}(R \leqslant 3)\) or using \({}^{20}C_4(p)^4(1 - p)^{16}\)
A1: awrt 0.0133
(ii) M1: writing or using \(1 - \mathrm{P}(R \leqslant 3)\)
A1: awrt 0.0159
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.05 \quad \mathrm{H}_1 : p \gt 0.05\) | B1 |
| \(\mathrm{P}(R \geqslant 4) = 1 - \mathrm{P}(R \leqslant 3)\) | M1 |
| \(= 0.2396 \qquad \text{CR } R \geqslant 8\) | A1 |
| Insufficient evidence to reject \(\mathrm{H}_0\), Not Significant. Accept \(\mathrm{H}_0\). 4 does not lie in the Critical region. | M1d |
| No evidence to support Patrick’s claim. Or no evidence that people in Reddman have a probability greater than 5% of having red hair | A1cso |
| (5) | |
| (12 marks) |
Notes
B1: Both hypotheses correct and labelled \(\mathrm{H}_0\) and \(\mathrm{H}_1\), must use \(p\) or \(\pi\). Do not allow \(p(x)\)
M1: Writing or using B(50, 0.05) AND writing or using \(1 - \mathrm{P}(R \leqslant 3)\) or \(\mathrm{P}(R \leqslant 3) = 0.7604\) on its own or one of the following 4 statements leading to a CR. \(\mathrm{P}(R \geqslant 7) = 0.0118\), \(\mathrm{P}(R \leqslant 6) = 0.9882\), \(\mathrm{P}(R \geqslant 8) = 0.0032\), \(\mathrm{P}(R \leqslant 7) = 0.9968\). May be implied by correct CR. Allow any letter
A1: awrt 0.240 or 0.24 or \(R \geqslant 8\) oe Or 0.7604
M1: dependent on the previous M being awarded. A correct statement – do not allow contradictory non contextual statements. Follow through their Probability/CR and \(\mathrm{H}_1\). If no \(\mathrm{H}_1\) seen then M0.
Ignore their comparison in all cases
Then mentally compare their probability as follows:
For prob < 0.5 statement must be correct compared to 0.01 for 1 tail test and 0.005 for 2 tailed test.
For prob > 0.5 statement must be correct compared to 0.99 for 1 tail test and 0.995 for 2 tailed test.
NB: If there is no non-contextual statement given you may award the M1 for a correct contextual statement
A1: cso fully correct solution and correct contextual statement containing the word Patrick if writing about the claim Or red hair if full context