- A machine produces components whose lengths are normally distributed with mean 102.3 mm and standard deviation 2.8 mm . After the machine had been serviced, a random sample of 20 components were tested to see if the mean and standard deviation had changed. The lengths, \(x \mathrm {~mm}\), of each of these 20 components are summarised as
$$\sum x = 2072 \quad \sum x ^ { 2 } = 214856$$
- Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a change in standard deviation.
- Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the mean length of the components has changed from the original value of 102.3 mm using
- a normal distribution,
- a \(t\) distribution.
- Comment on the mean length of components produced after the service in the light of the tests from part (a) and part (b). Give a reason for your answer.