Find the critical values $\chi^{2}$ and $\chi^{2}{ }_{R}$ for the given confidence level $c$ and sample size $n$. \[ \mathrm{c}=0.9, \mathrm{n}=23 \] $\chi_{L}^{2}=\square$ (Round to three decimal places as needed.)

Step 1 ：We are given a confidence level of c=0.9 and a sample size of n=23. We are asked to find the critical value \(\chi_{L}^{2}\) for these parameters.

Step 2 ：The critical value \(\chi_{L}^{2}\) corresponds to the lower tail of the chi-square distribution. For a confidence level of c=0.9, the lower tail covers (1-c)/2 = 0.05 of the distribution.

Step 3 ：Therefore, we need to find the chi-square value such that the cumulative probability up to that value is 0.05.

Step 4 ：The chi-square distribution is defined for positive values and has two parameters: degrees of freedom and non-centrality parameter. In this case, the degrees of freedom is n-1, and the non-centrality parameter is 0 (for a central chi-square distribution).

Step 5 ：So, the degrees of freedom df = n - 1 = 23 - 1 = 22.

Step 6 ：Using these parameters, we can calculate the critical value \(\chi_{L}^{2}\).

Step 7 ：The calculated critical value \(\chi_{L}^{2}\) is approximately 12.338.

Step 8 ：Final Answer: The critical value \(\chi_{L}^{2}\) for the given confidence level c=0.9 and sample size n=23 is \(\boxed{12.338}\).