USA: +1-585-535-1023

UK: +44-208-133-5697

AUS: +61-280-07-5697

Testing of a Single Variance

Consider a normal population N (µ, σ2).

1. Set up H0 : σ2

2. Set up H1 : σ2 < σ02 or σ2 > σ02,  or σ2 ≠ σ02

3. Test statistic:

S2 = 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 X2tab using chi-square table with

(n-1) degrees of freedom.

5. Compute the statistic, say X2cab.

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 S2 = 0.000324.

Solution.         1. H0 : σ = 0.022 inch.

2. H1 : σ ≠ 0.022 inch.

3. Test statistic: 

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      X20.975,1.7 = 7.564 and

 

X20.025 , 17 = 30.191.

5. Computation

6. Decision:

 

Since Xcal < 30.191 and X2cal > 7.564

it lies in the acceptance region.

=>        H0  is accepted.

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