A flywheel is retained on a shaft by five bolts, which are each tightened to a specified torque of 50 ± 5 Nm. A sample of 20 assemblies was checked for bolt torque. The results from the 100 bolts had a mean of 47.2Nm and a standard deviation of 1.38 Nm.

a. Assuming that torques are normally distributed, estimate the proportion below 45 Nm.

b. For a given assembly, what is the probability of

(i) there being no bolts below 45 Nm;

(ii) there being at least one bolt below 45 Nm;

(iii) there being fewer than two bolts above 45 Nm; (iv) all five bolts being below 45 Nm.

c. In the overall sample of 100 bolts, four were actually found with torques below 45 Nm.

(i) Comment on the comparison between this result and your answer to (a) above.

(ii) Use this result to obtain a 90% two-sided confidence interval for the proportion below 45 Nm.

d. Explain the meaning of the confidence interval in c (ii) above as you would to an intelligent, but non-technically-minded, manager.

e. The lowest torque bolt in each assembly was identified. For these 20 bolts, the mean torque was 45.5Nm and the standard deviation 0.88 Nm. Assuming an appropriate extreme-value distribution, calculate the probability thaton a given assemblythe lowest torque will be

(i) below 45 Nm;

(ii) below 44 Nm.