4   2. A manufacturer uses a machine to make metal rods. The length of a metal rod, L cm, is normally distributed with • a mean of 8 cm • a standard deviation of x cm Given that the proportion of metal rods less than 7.902 cm in length is 2.5% (a) show that x = 0.05 to 2 decimal places. (2) (b) Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. (1) The cost of producing a single metal rod is 20p A metal rod • where L < 7.94 is sold for scrap for 5p • where 7.94  L  8.09 is sold for 50p • where L > 8.09 is shortened for an extra cost of 10p and then sold for 50p (c) Calculate the expected profit per 500 of the metal rods. Give your answer to the nearest pound. (5) The same manufacturer makes metal hinges in large batches. The hinges each have a probability of 0.015 of having a fault. A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95% of batches to be accepted. (d) Explain whether the manufacturer is likely to achieve its aim. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 5   Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________