Question

find the critical values $chi ^{2}_{l}$ and $chi ^{2}_{r}$ for the given confidence level $c$ and sample size $n$.
$c = 0.9$, $n = 30$
$chi ^{2}_{l}=17.708$ (round to three decimal places as needed.)
$chi ^{2}_{r}=square$ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 30\), so \(df=30-1=29\).

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 a \(\chi^{2}\) - distribution table or a calculator with a \(\chi^{2}\) - distribution function (e.g., in Excel, use the formula \(=\text{CHISQ.INV.RT}(0.05,29)\)), we find the value.

Answer:

\(\chi_{R}^{2}=42.557\)