S2 January 2011 Q1
1. A disease occurs in 3% of a population.
A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
| Scheme | Marks |
|---|---|
| Occurrences of the disease are independent The probability of catching the disease remains constant. | B1 B1 |
| (2) |
Notes
B1 independent
B1 probability remains constant.
One of these must have the context of disease.
No context only one correct B0B0
If only one mark awarded give the first B1
SC if they are both correct without context award B1B0
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{Bin}(10, 0.03)\) | B1 |
| \(\mathrm{P}(X = 2) = \dfrac{10 \times 9}{2}(0.03)^2(0.97)^8 = 0.0317\) | M1A1 |
| (3) |
Notes
B1 for writing or using B(10,0.03)
M1 for writing or using \((p)^2(1 - p)^8\dfrac{10!}{2!8!}\) allow \({}^{10}\mathrm{C}_2, \dbinom{10}{2}\) etc
Allow \(\mathrm{P}(X \leqslant 2) - \mathrm{P}(X \leqslant 1)\)
A1 awrt 0.0317
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = 100 \times 0.03 = 3\) | B1cao |
| \(\mathrm{Var}(X) = 100 \times 0.03 \times 0.97 = 2.91\) | B1cao |
| (2) |
| Scheme | Marks |
|---|---|
| \(\lambda = 100 \times 0.03 = 3\) \(Y \sim \mathrm{Po}(3)\) | B1 (use of) |
| \(\mathrm{P}(Y \gt 5) = 1 - \mathrm{P}(Y \leqslant 5)\) | dM1 |
| \(= 1 - 0.9161\) \(= 0.0839\) | A1 |
| (3) | |
| (10 marks) |
Notes
B1 for using Poisson. Any mean. Common values which imply Poisson used are 0.9665 and 0.8153
dM1 for writing or using \(1 - \mathrm{P}(X \leqslant 5)\) - use of binomial gets M0.
This is dependent on them being awarded the previous B mark.
A1 awrt 0.0839
SC: Use of Normal in (d)
Can get B0 M1 A0.- for M1 we must see \(1 - \mathrm{P}(X \leqslant 5)\) or \(1 - \mathrm{P}(X \leqslant 5.5)\) oe or get awrt 0.071