Compare tests before and after pooling two samples

A manufacturing process produces ball bearings with diameters with known standard deviation 0.04 cm. Under normal circumstances, the manufacturing process will produce ball bearings of mean diameter 0.5 cm.

(i) During a routine quality control check, a random sample of 25 ball bearings gives a mean of 0.51 cm. Is there evidence to believe that the manufacturing process is producing ball bearings of the stated diameter? Perform an appropriate test at 5% level of significance. State a necessary assumption for the test to be valid.4 marks

An enhancement on the manufacturing process will ensure that the diameters of the ball bearings produced are less variable.

(ii) Measurements of a sample of 100 ball bearings give the following summary statistics:

$\sum x=50.6,\qquad\sum(x-0.5)^2=0.08345.$

Show that the unbiased estimate of the population variance is $8.07\times 10^{-4}$.

Is there evidence at the 5% level of significance that after the enhancement, the manufacturing process is producing oversized ball bearings?4 marks

(iii) Another sample of 100 ball bearings yield the same summary statistics as the previous sample in (ii). Explain, with justification, whether the combined sample will give a different conclusion to (ii).4 marks