- Two independent random samples \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated.
$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$
Find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on these two samples.