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 $\boxed{{\displaystyle 32.671}}$ and $\boxed{{\displaystyle 11.591}}$.