Question

find the critical values $\chi_{r}^{2}$ and $\chi_{l}^{2}$ for the given confidence level c and sample size n.
c = 0.95, n = 20
$\chi_{r}^{2}=\square$ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 20\), so \(df=20-1 = 19\).

Step2: Find the right - tailed critical value

For a confidence level \(c = 0.95\), the significance level \(\alpha=1 - c=1 - 0.95 = 0.05\). The right - tailed area is \(\frac{\alpha}{2}=0.025\).
Using the chi - square distribution table or a calculator with a chi - square function (\(\chi^{2}\) distribution), for \(df = 19\) and right - tailed area \(0.025\), we get \(\chi_{R}^{2}=\chi_{0.025}^{2}(19)\).
Looking up in the chi - square table or using a calculator (e.g., in R: qchisq(0.975,19)), we find \(\chi_{R}^{2}=32.852\).

Answer:

\(32.852\)