4. Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x \mathrm {~cm}\), of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below.
| Sample size | Mean \(\bar { x }\) | |
| Machine \(A\) | 9 | 4.83 | 0.721 |
| Machine \(B\) | 10 | 4.85 | 0.572 |
The lengths of components produced by the machines can be assumed to follow normal distributions.
- Use a two tail test to show, at the \(10 \%\) significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal.
(4) - Showing your working clearly, find a \(95 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively.
There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm .
- State, giving your reason, whether or not the factory manager should be concerned.