Question

suppose you are given the statistics ( n = 43 ) and ( s ^ { 2 } = 500 ) and wish to test
( h _ { 0 } : sigma ^ { 2 } = 200 )
( h _ { a } : sigma ^ { 2 }
eq 200 )
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.
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Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom for a chi - square test of variance is \(df=n - 1\). Given \(n = 43\), then \(df=43-1 = 42\).

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

For a two - tailed test with \(\alpha=0.05\), the area in each tail is \(\frac{\alpha}{2}=0.025\).
Looking up in the chi - square distribution table:
The lower - tail critical value \(\chi_{1-\frac{\alpha}{2},df}^2=\chi_{0.975,42}^2\approx25.695\)
The upper - tail critical value \(\chi_{\frac{\alpha}{2},df}^2=\chi_{0.025,42}^2\approx60.481\)

Answer:

\(25.695,60.481\)