A country baker makes biscuits whose masses are normally distributed with mean 30 g and standard deviation 2.3 g. She packs them by hand into either a small carton (containing 20 biscuits) or a large carton (containing 30 biscuits).

(a) State the distribution of the total mass, S, of biscuits in a small carton and find the probability that S is greater than 615 g.

(b) Six small and four large cartons are placed in a box. Find the probability that the total mass of biscuits in the 10 cartons lies between 7150 g and 7250 g.

(c) Find the probability that 3 small cartons. contain at least 25 g more than 2 large ones. The label on a large carton of biscuits reads 'Net mass 900 g. A trading standards officer insists that 90% of such cartons should contain biscuits with a total mass of at least 900 g.

(d) Assuming the standard deviation remains unchanged, find the least value of the mean mass of a biscuit consistent with this requirement. (MEI)