The lengths, in centimetres, of rods produced in a factory have mean µ and population standard deviation 0.2. The value of µ is supposed to be 250, but a manager claims that one machine is producing rods that are too long on average. 

(i) A random sample of 40 rods from the above machine is taken and the sample mean length is found to be 250.06 cm. Test at the 5% significance level whether the manager’s claim is justified.             

(ii) If the population standard deviation was unknown and the sample standard deviation was 0.6, would the test change? Explain?

(iii) Test for the validity of population standard deviation figure given in (iii).