4. The length \(X \mathrm {~mm}\) of a spring made by a machine is normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 20 springs is selected and their lengths measured in mm . Using this sample the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) are
$$\bar { x } = 100.6 , \quad s ^ { 2 } = 1.5 .$$
Stating your hypotheses clearly test, at the \(10 \%\) level of significance,
- whether or not the variance of the lengths of springs is different from 0.9 ,
- whether or not the mean length of the springs is greater than 100 mm .