The lengths of pencils made at a factory are normally distributed. The standard deviation of the lengths is \(\sigma\) cm, and the mean is supposed to be 10 cm. An inspector thinks that the mean is actually greater than 10 cm. He takes a random sample of 50 pencils produced at the factory and finds that the mean of these 50 lengths is 10.03 cm. He then carries out a hypothesis test.

(a) He finds that the value of the test statistic \(z\) is 1.995 correct to 3 decimal places.

(a(i)) Calculate the value of \(\sigma\). [3]

(a(ii)) Carry out the hypothesis test at the 2.5% significance level. [3]

(b) Explain whether it was necessary to use the Central Limit Theorem in carrying out the test. [1]