Find the critical values \( \chi_{L}^{2} \) and \( \chi_{R}^{2} \) for the given confidence level \( c \) and sample size \( n \). \[ =0.95, n=20 \] \( \chi_{L}^{2}=\square \) (Round to three decimal places as needed.)

To find the critical values \( \chi_{L}^{2} \) and \( \chi_{R}^{2} \) for a given confidence level \( c = 0.95 \) and sample size \( n = 20 \), we first need to determine the degrees of freedom (df) for the chi-squared distribution. The degrees of freedom is calculated as: \[ \text{df} = n - 1 = 20 - 1 = 19 \] For a confidence level of \( 0.95 \), we need to find the critical values that correspond to the lower and upper tails of the chi-squared distribution. Since the confidence level is \( 0.95 \), the significance level \( \alpha \) is: \[ \alpha = 1 - c = 1 - 0.95 = 0.05 \] This means that we will have \( \alpha/2 = 0.025 \) in each tail of the distribution. Now, we can find the critical values using a chi-squared distribution table or a calculator: 1. **Lower critical value \( \chi_{L}^{2} \)** corresponds to the \( 0.025 \) quantile (left tail). 2. **Upper critical value \( \chi_{R}^{2} \)** corresponds to the \( 0.975 \) quantile (right tail). Using a chi-squared distribution table or calculator for \( df = 19 \): - For \( \chi_{L}^{2} \) (0.025 quantile): - \( \chi_{L}^{2}(0.025, 19) \approx 10.117 \) - For \( \chi_{R}^{2} \) (0.975 quantile): - \( \chi_{R}^{2}(0.975, 19) \approx 30.144 \) Thus, the critical values are: \[ \chi_{L}^{2} \approx 10.117 \quad \text{(rounded to three decimal places: } 10.117\text{)} \] \[ \chi_{R}^{2} \approx 30.144 \quad \text{(rounded to three decimal places: } 30.144\text{)} \] So, the final answer for \( \chi_{L}^{2} \) is: \[ \chi_{L}^{2} = 10.117 \]