Question

find the critical values ( chi_{1 - alpha/2}^2 ) and ( chi_{alpha/2}^2 ) for a 80% confidence level and a sample size of ( n = 10 ).
( chi_{1 - alpha/2}^2=square )
(round to three decimal places as needed.)

Explanation:

Step1: Determine the significance level $\alpha$

The confidence level is $80\%$, so $\alpha = 1 - 0.80=0.20$. Then $\frac{\alpha}{2}=0.10$ and $1 - \frac{\alpha}{2}=0.90$.

Step2: Find the degrees of freedom

The sample size is $n = 10$. The degrees of freedom is $df=n - 1=10 - 1 = 9$.

Step3: Look up the critical values in the chi - square distribution table

For $\chi^{2}_{1-\alpha/2}$ with $df = 9$ and $1-\alpha/2 = 0.90$, from the chi - square distribution table, $\chi^{2}_{0.90,9}=4.168$.
For $\chi^{2}_{\alpha/2}$ with $df = 9$ and $\alpha/2 = 0.10$, from the chi - square distribution table, $\chi^{2}_{0.10,9}=14.684$.

Answer:

$\chi^{2}_{1-\alpha/2}=4.168$ and $\chi^{2}_{\alpha/2}=14.684$