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
$chi_{r}^{2}=32.852$ (round to three decimal places as needed.)
$chi_{l}^{2}=$ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the significance level and degrees of freedom

The significance level \(\alpha = 1 - c\). Given \(c = 0.95\), so \(\alpha=1 - 0.95=0.05\).
The degrees of freedom \(df=n - 1\). Given \(n = 20\), so \(df=20 - 1 = 19\).
For the left - tailed critical value \(\chi_{L}^{2}\), the area to the left of \(\chi_{L}^{2}\) is \(\frac{\alpha}{2}=0.025\).

Step2: Use the chi - square distribution table or calculator

Using a chi - square distribution table or a calculator with a chi - square distribution function (e.g., in Excel, use the function CHISQ.INV(0.025,19) or in R use qchisq(0.025,19)).

$$ \chi_{L}^{2}=8.907 $$

Answer:

\(\chi_{L}^{2}=8.907\)