S2 June 2005 Q1
1. It is estimated that 4% of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5.
(a) Calculate the value of \(n\). (3)
The expected number of people with green eyes in a second random sample is 3.
(b) Find the standard deviation of the number of people with green eyes in this second sample. (4)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(n, 0.04)\) | B1 |
| \(\mathrm{E}(X) = np\) \(5 = 0.04n\) | M1 |
| \(n = 125\) | A1 |
| (3) |
Notes
B1 implied
M1 use of \(np = 5\)
A1 125
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = 3\) \(np = 3\) | B1 |
| \(\text{sd} = \sqrt{npq} = \sqrt{3(1 - 0.04)}\) | M1 |
| \(= \sqrt{2.88}\) | A1 |
| \(= 1.70\) | A1 |
| (4) | |
| (7 marks) |
Notes
B1 \(np = 3\)
M1 use of \(\sqrt{npq}\)
1st A1 \(\sqrt{3(1 - 0.04)}\)
2nd A1 awrt 1.70