Question

suppose you are given the statistics ( n = 49 ) and ( s ^ { 2 } = 700 ) and wish to test ( h _ { 0 } : sigma ^ { 2 } = 500 ) ( h _ { a } : sigma ^ { 2 }
eq 500 ) using a significance level of ( 5 % ). what critical value(s) from the chi - square distribution table would be used in this test? if there is more than one value, enter them as a comma - separated list. click here to expand the distribution tables.

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 49\), then \(df=49-1 = 48\).

Step2: Find the critical values for a two - tailed test

For a two - tailed test with a significance level of \(\alpha=0.05\), the area in each tail is \(\frac{\alpha}{2}=0.025\).
Looking up in the chi - square distribution table:
The lower critical value \(\chi_{1-\frac{\alpha}{2},df}^2=\chi_{0.975,48}^2\approx30.755\)
The upper critical value \(\chi_{\frac{\alpha}{2},df}^2=\chi_{0.025,48}^2\approx68.157\)

Answer:

\(30.755,68.157\)