(a) It is estimated on average that one match in five in the Football League is a draw. And that one match in two is a home win.

(i) A random sample of 12 matches are selected. Find the probability that the number of draw matches is:

a) Exactly 3.

b) At least 4. [4 marks]

(ii) A random sample of 90 matches are selected. Use a suitable approximation to find the probability of draw matches that are between 13 and 20 (inclusive) [5 marks]

(iii) A random sample of 20 matches are selected. The random variables D and H are the numbers of draw matches and home wins respectively. State, with a reason, which one between D and H is better approximated by a normal variable. [3 marks]

(b) Given that represent the random variable "the breaking strength of paving stones laid in public areas" and is normally distributed with mean 50 units and standard deviation 8 units. Random samples of paving stones are taken. The mean breaking strength for a sample is denoted by . (i) State the distribution of , giving its mean and variance.

[1 mark]

(ii) For a sample size of = 25, find the probability that exceeds 54 units. [2 marks]

iii) Find the smallest possible sample size if the probability that exceeds 54 units is less than 0.01. [3 marks]

Suppose that the breaking strength of paving stones laid in public areas has mean 50 units and standard deviation 8 units, but that the form of the distribution of breaking strengths is not known. Random samples of paving stones are taken. Discuss about the form of the distribution of the mean breaking strength of these samples (iv)in the case when is large, (v) and also in the case when is small?