A batch of 100 steel rods passes inspection if the average of their diameters falls between 0.495 cm and 0.505 cm. Let µ and σ denote the mean and standard deviation, respectively, of the diameter of...

A batch of 100 steel rods passes inspection if the average of their diameters falls between 0.495 cm and 0.505 cm. Let µ and σ denote the mean and standard deviation, respectively, of the diameter of a randomly selected rod. Answer the following questions assuming that µ = 0.503 cm and σ = 0.03 cm.

(a) What is the (approximate) probability the inspector will accept (pass) the batch?

(b) Over the next 6 months 40 batches of 100 will be delivered. Let X denote the number of batches that will pass inspection.

(i) State the exact distribution of X, and use R to find the probability P(X ≤ 30).

(ii) Use the DeMoivre-Laplace Theorem, with and without continuity correction, to approximate P(X ≤ 30). Comment on the quality of the approximation provided by the two methods.