Question

find the $\chi^{2}$ critical values for a 90% confidence interval and 24 degrees of freedom. round answers to 3 decimal places.
confidence level $\square\\%$ and $\alpha=\square$, also $\frac{\alpha}{2}=\square$, and $1 - \frac{\alpha}{2}=\square$
critical values: lower=$\square$ upper=$\square$

Explanation:

Step1: Calculate confidence level and $\alpha$

Given confidence level is \(90\%\). So, \(\alpha=1 - 0.90=0.10\)

Step2: Calculate \(\frac{\alpha}{2}\) and \(1-\frac{\alpha}{2}\)

\(\frac{\alpha}{2}=\frac{0.10}{2}=0.05\) and \(1-\frac{\alpha}{2}=1 - 0.05=0.95\)

Step3: Find lower critical value

Using \(\chi^{2}\) - distribution table or calculator, for \(\frac{\alpha}{2}=0.05\) and \(df = 24\), the lower critical value \(\chi_{1-\frac{\alpha}{2},df}^{2}=\chi_{0.95,24}^{2}\)
From \(\chi^{2}\) - table or calculator: \(\chi_{0.95,24}^{2}=13.848\)

Step4: Find upper critical value

For \(\frac{\alpha}{2}=0.05\) and \(df = 24\), the upper critical value \(\chi_{\frac{\alpha}{2},df}^{2}=\chi_{0.05,24}^{2}\)
From \(\chi^{2}\) - table or calculator: \(\chi_{0.05,24}^{2}=36.415\)

Answer:

Confidence level \(90\%\) and \(\alpha = 0.10\), also \(\frac{\alpha}{2}=0.05\), and \(1-\frac{\alpha}{2}=0.95\)
Critical values: lower \(13.848\) upper \(36.415\)