S2 January 2006 Q6
6. A bag contains a large number of coins. Half of them are 1p coins, one third are 2p coins and the remainder are 5p coins.
(a) Find the mean and variance of the value of the coins. (4)
A random sample of 2 coins is chosen from the bag.
(b) List all the possible samples that can be drawn. (3)
(c) Find the sampling distribution of the mean value of these samples. (6)
| Scheme | Marks | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| |||||||||
| \(\text{Mean} = 1 \times \dfrac{1}{2} + 2 \times \dfrac{1}{3} + 5 \times \dfrac{1}{6} = 2 \quad\) or 0.02 | M1A1 | ||||||||
| \(\text{Variance} = 1^2 \times \dfrac{1}{2} + 2^2 \times \dfrac{1}{3} + 5^2 \times \dfrac{1}{6} - 2^2 = 2 \quad\) or 0.0002 | M1A1 | ||||||||
| (4) |
Notes
1st M1 \(\Sigma x.p(x)\) need \(\frac{1}{2}\) and \(\frac{1}{3}\)
2nd M1 For M \(\Sigma x^2.p(x) - \lambda^2\)
| Scheme | Marks |
|---|---|
| (1,1) (1,2) and (2,1) | B2 |
| (1,5) and (5,1) | B1 |
| e.e. (2,2) (2,5) and (5,2) (5,5) | B1 |
| (3) |
Notes
LHS −1 (e.e.)
repeat of “theirs” on RHS
NB the marks printed in this part (B2, B1, B1) add to 4, but the part total is printed as (3) and the question total as 13.
| Scheme | Marks | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1A1 M1A2 | ||||||||||||||
| (6) | |||||||||||||||
| (13 marks) |
Notes
M1A1 \(\frac{1}{4}\)
M1A2 1.5 +, −1 ee