QUESTION 2 (a) Two boxes contain three cards each. The first box contains cards labelled 1,3, and 5; the second box contains cards labelled 2, 6, and 8. In a game, a player draws one card at random from each box and his score, X, is the sum of the numbers on the two cards.

(i) Obtain the SIX (6) possible values of X and find the corresponding probabilities (3 marks)

(ii) Calculate E(X) and variance of X. (5 marks)

(b) A study has shown that 70% of the families in a certain town have at least two cars. Find the probabilities that among 10 families randomly selected in this town,

(i) there are 8 families have at least two cars.

(2 marks)

(ii) 5 families have fewer than two cars. (3 marks)

(c) Company records show that the weekly distance travelled by their salesmen is approximately normally distributed with mean 800 miles and standard deviation 90 miles. The sales manager considers that salesmen who travel less than 600 miles in one week are performing poorly.

(i) If the company employs 200 salesmen, how many would be expected to perform poorly in a particular week? (4 marks)

(ii) The sales manager wishes to identify the number of miles travelled in one week, above which only 1% of salesmen are expected to exceed. What weekly mileage is this? Round your answer to the nearest miles. (3 marks)

(d) A factory produces butter with nominal weight of 125 g each. From experience, it is known that 10% of the butter have weights less than the nominal weight and 5% of the butter have weights exceeding 130g. Assuming that the weight of the butter produced by the factory is distributed normally, find the mean and standard deviation of the weight of the butter. (5 marks)