Use the given information to find the number of degrees of freedom, the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$, and the confidence interval estimate of $\sigma$. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution.
Nicotine in menthol cigarettes $98 \%$ confidence; $n=21, \mathrm{~s}=0.26 \mathrm{mg}$.
ت Click the icon to view the table of Chi-Square critical values.
$\mathrm{df}=20$ (Type a whole number.)
$\chi_{L}^{2}=\square$ (Round to three decimal places as needed.)

Step 1 ：The degrees of freedom is calculated by subtracting 1 from the sample size. In this case, the sample size is 21, so the degrees of freedom is \(21 - 1 = 20\).

Step 2 ：The critical values \(\chi_{L}^{2}\) and \(\chi_{R}^{2}\) are found using the chi-square distribution table for the given confidence level and degrees of freedom. The confidence level is 98%, so the significance level is \(100\% - 98\% = 2\%\). This is split between the two tails of the distribution, so we look up the critical values for 1% in each tail. The critical values are \(\chi_{L}^{2} = 8.260\) and \(\chi_{R}^{2} = 37.566\).

Step 3 ：The confidence interval estimate of \(\sigma\) is calculated using the formula \(\sqrt{\frac{(n-1)s^2}{\chi^2}}\) for the lower bound and \(\sqrt{\frac{(n-1)s^2}{\chi^2}}\) for the upper bound, where s is the sample standard deviation, n is the sample size, and \(\chi^2\) is the chi-square critical value. In this case, the lower bound is \(0.036\) and the upper bound is \(0.164\).

Step 4 ：Final Answer: The degrees of freedom is \(\boxed{20}\). The critical values are \(\chi_{L}^{2} = \boxed{8.260}\) and \(\chi_{R}^{2} = \boxed{37.566}\). The confidence interval estimate of \(\sigma\) is \(\boxed{(0.036, 0.164)}\).