Determine the critical values for the confidence interval for the population standard deviation from the given values. Round your answers to three decimal places. \[ n=22 \text { and } \alpha=0.1 \]

Step 1 ：The problem requires finding the critical values for the confidence interval for the population standard deviation. This is done using the chi-square distribution.

Step 2 ：The critical values are given by the chi-square distribution with \(n-1\) degrees of freedom. The chi-square distribution is not symmetric, so the critical values are not symmetric around the mean. The critical values are found at the \(\alpha/2\) and \(1-\alpha/2\) quantiles of the chi-square distribution.

Step 3 ：The degrees of freedom (df) is calculated as \(n-1 = 22 - 1 = 21\).

Step 4 ：The alpha level (\(\alpha\)) is 0.1, so \(\alpha/2 = 0.05\) and \(1-\alpha/2 = 0.95\).

Step 5 ：Using a chi-square table or calculator, we find the critical values at these quantiles.

Step 6 ：The chi-square value for df=21 and \(\alpha/2=0.05\) is approximately 32.671.

Step 7 ：The chi-square value for df=21 and \(1-\alpha/2=0.95\) is approximately 11.591.

Step 8 ：So, the critical values for the confidence interval for the population standard deviation are approximately 32.671 and 11.591.

Step 9 ：Therefore, the final answer is \(\boxed{32.671}\) and \(\boxed{11.591}\).