Question

find the critical values $\chi^{2}_{l}$ and $\chi^{2}_{r}$ for the given confidence level c and sample size n.
$c = 0.9$, $n = 21$
$\chi^{2}_{l}=10.851$ (round to three decimal places as needed.)
$\chi^{2}_{r}=\square$ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the degree of freedom

The degree of freedom \(df=n - 1\). Given \(n = 21\), so \(df=21-1=20\).

Step2: Calculate the right - tail area

The confidence level \(c = 0.9\), so the significance level \(\alpha=1 - c=1 - 0.9 = 0.1\). The right - tail area is \(\frac{\alpha}{2}=0.05\).

Step3: Find the critical value \(\chi_{R}^{2}\)

Using the chi - square distribution table or a calculator with a chi - square distribution function, for \(df = 20\) and right - tail area \(0.05\), we get \(\chi_{R}^{2}=\chi_{0.05,20}^{2}\).
From the chi - square distribution table or calculator, \(\chi_{0.05,20}^{2}=31.410\).

Answer:

\(31.410\)