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
$chi^{2}_{l}=square$ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the significance level $\alpha$

The confidence level is \(c = 0.9\). The significance level is \(\alpha=1 - c\). So, \(\alpha=1 - 0.9=0.1\).

Step2: Calculate the degrees of freedom \(df\)

The sample size is \(n = 21\). The degrees of freedom for a chi - square distribution in this context is \(df=n - 1\). So, \(df=21 - 1 = 20\).

Step3: Find \(\chi_{L}^{2}\)

For the left - tailed critical value \(\chi_{L}^{2}\), we use the value \(\alpha/2=0.1/2 = 0.05\) and \(df = 20\). Looking up in the chi - square distribution table (or using a calculator with a chi - square inverse function, e.g., in Excel: =CHISQ.INV(0.05,20)), we get \(\chi_{L}^{2}=10.851\).

Answer:

\(10.851\)