- Two machines, \(A\) and \(B\), are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.
Samples are taken from each machine and the amount of water, \(x \mathrm { ml }\), dispensed in each bottle is recorded. The table shows the summary statistics for Machine \(A\).
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | Sample size | \(\sum x\) | \(\sum x ^ { 2 }\) |
| Machine \(A\) | 9 | 2268 | 571700 |
- Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine \(A\).
For Machine \(B\), a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is \(12.7 \mathrm { ml } ^ { 2 }\)
- Test, at the \(10 \%\) level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.