Consider a normal population N (µ, σ^{2}).

1. Set up H_{0} : σ^{2}

2. Set up H_{1} : σ^{2} < σ_{0}^{2} or σ^{2} > σ_{0}^{2}, or σ^{2 ≠ }σ_{0}^{2}

S^{2} = Sample variance (unbiased)

which follows chi-square distribution with (n - 1) degrees of freedom

4. Set up the level of significance α and the critical point as X^{2}_{tab} using chi-square table with

(n-1) degrees of freedom.

5. Compute the statistic, say X^{2}_{cab.}

Example 12. Use the 0.05 level of significance to test the null hypothesis that σ = 0.022 inch for the diameters of certain wire rope against the alternative hypothesis that σ ≠ 0. 022 inch, given that a random sample of size 18 yielded S^{2} = 0.000324.

Solution. 1. H_{0} : σ = 0.022 inch.

2. H_{1} : σ ≠ 0.022 inch.

which follows chi-square distribution with (n - 1) degrees of freedom.

4. Here α = 0.05. Alternative hypothesis shows that it is right tailed test. α/2 = 0.025.

Critical values are X^{2}_{0.975},_{1.7 }= 7.564 and

X^{2}_{0.025 }, _{17} = 30.191.

5. Computation

6. Decision:

Since X_{cal }< 30.191 and X^{2}_{cal} > 7.564

it lies in the acceptance region.

=> H_{0 }is accepted.

=> True diameter of the metallic rope can be 0.022 inch.