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
$\chi^{2}_{l}=$ (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 significance level \(\alpha\)

The confidence level \(c = 0.9\), then \(\alpha=1 - c=1 - 0.9=0.1\).

Step3: Calculate \(\frac{\alpha}{2}\)

\(\frac{\alpha}{2}=\frac{0.1}{2}=0.05\).

Step4: Find \(\chi_{L}^{2}\) using the chi - square distribution table or a calculator

For a left - tailed critical value \(\chi_{L}^{2}\) with \(df = 29\) and area to the left of \(\chi_{L}^{2}\) equal to \(\frac{\alpha}{2}=0.05\). Using a chi - square distribution table or a calculator (e.g., in R: qchisq(0.05,29) or in Excel: CHISQ.INV(0.05,29)), we get \(\chi_{L}^{2}=17.708\).

Answer:

\(17.708\)