Question: An engineer was interested in comparing the variability of the power of a laser at current levels of 16 amps \((X)\) and 20 amps (Y). To do so, random samples of 25 observations of power readings were obtained at each current level. The samples were taken in such a manner that independence could be assumed.

Sample variances of $0.012$ and $0.020$ were obtained for \(X\) and \(Y\), respectively. That is, the sample observations resulted in \(\mathrm{s}_{\mathrm{X}}^{2}=0.012\) and \(\mathrm{sy}^{2}=0.020\). Also, the data indicated that the sampled populations were normally distributed.

1. Find a \(95 \%\) confidence interval for the variance of \(\mathrm{X}, \sigma_{\mathrm{X}}^{2}\).
2. Find a \(95 \%\) confidence interval for the variance of \(Y, \sigma_{Y}^{2}\).
3. Find a \(95 \%\) confidence interval for the ratio of the variances of \(\mathrm{X}\) and \(\mathrm{Y}, \sigma_{\mathrm{X}}^{2} / \sigma_{\mathrm{Y}}^{2} .\)
4. Can you conclude that the respective variances of power at each of the two current levels may be the same? Are your results for parts \(\mathrm{a}, \mathrm{b}\) and c consistent. Explain your answer fully.

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