Find the critical values $\chi_{R}^{2}$ and $\chi_{L}^{2}$ for the given confidence level $c$ and sample size $n$. \[ \mathrm{c}=0.80, \mathrm{n}=20 \] $\chi_{\mathrm{R}}^{2}=\square($ Round to three decimal places as needed.)

Step 1 ：We are given a confidence level \(c = 0.80\) and a sample size \(n = 20\).

Step 2 ：The degrees of freedom for a chi-square distribution is \(n-1\), where \(n\) is the sample size. So, the degrees of freedom \(df = 19\).

Step 3 ：The right critical value \(\chi_{R}^{2}\) corresponds to the upper tail of the distribution, and the left critical value \(\chi_{L}^{2}\) corresponds to the lower tail.

Step 4 ：The confidence level \(c\) is the area under the curve between these two critical values. Therefore, the area in the upper tail is \((1-c)/2\) and the area in the lower tail is also \((1-c)/2\).

Step 5 ：Using the chi-square percent point function (ppf), we find the critical values. The first argument to this function is the cumulative probability (i.e., the area under the curve to the left of the critical value), and the second argument is the degrees of freedom.

Step 6 ：By calculation, we find that \(\chi_{R}^{2} = 27.203571029356844\) and \(\chi_{L}^{2} = 11.650910032126951\).

Step 7 ：Rounding to three decimal places, we get \(\chi_{R}^{2} = 27.204\) and \(\chi_{L}^{2} = 11.651\).

Step 8 ：Final Answer: The critical values for the given confidence level \(c=0.80\) and sample size \(n=20\) are \(\boxed{\chi_{R}^{2} = 27.204}\) and \(\boxed{\chi_{L}^{2} = 11.651}\).