Problem

(a) Explain what you understand by the sampling distribution of a statistic.

A bag contains a large number of £1 coins and a large number of £2 coins only.

A random sample of 3 coins is taken from the bag.

(b) List all the possible combinations of the 3 coins that might be selected.

The probability of selecting a £1 coin from the bag is $p$ and the probability of selecting a £2 coin from the bag is $q$.

Let $\bar{X}$ represent the mean value of the 3 coins selected.

The sampling distribution of $\bar{X}$ is given below.

$\bar{x}$	$\dfrac{4}{3}$	1	$a$	2
$P(\bar{X} = \bar{x})$	$\dfrac{27}{125}$	$b$	$c$	$d$

where $a$, $b$, $c$ and $d$ are constants.

(c) State the value of $a$.

(d) Find the value of $b$, the value of $c$ and the value of $d$.

Let $M$ represent the modal value of the 3 coins selected.

(e) Find the sampling distribution of $M$.

(Total for Question 3 is 13 marks)