A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of corks is represented by a Normal distribution with mean 3cm and standard deviation 0.15 cm. The specifications require corks with diameters between 2.85cm and 3.15cm. A cork that does not meet the specifications is considered defective (a cork that is too small leaks and causes the wine to deteriorate; a cork that is too large does not fit in the bottle). (a) What proportion of corks produced by this machine are defective? 2 MARKS (b) Ifyouchoose15corksatrandomproducedbythismachine,whatistheprobability that at most 5 of them (that is, 5 or fewer) will be defective? 3 MARKS (c) What machine standard deviation would be required if the manufacturer insists that the probability that a cork is defective must be no more than 0.02? 3 MARKS Justify your answer and show your working in each case