Consider normal population, N (µ, σ2)

• Set up H0 : σ2
• Set up H1 : σ202 or σ202, or σ2≠ σ02
• Statistics:

X2 =

Here,

S2 = Sample Variance (Unbiased)

It follows the chi-square distribution with the (n – 1) degree of freedom.

• Set up level of significance α, critical point, X2tab using the chi-square table along with (n – 1) degree of freedom.
• Computing statistics i.e., X2cal
 H1 Reject H0, if σ2< σ02 X2cal σ02 X2cal> X2tab i.e., X2α, n -1 σ2 ≠ σ02 X2cal< X2tab i.e., X21 – α/ 2, n -1or,X2cal> X2tab i.e., X2α/ 2, n -1

Example No. 12:

Suppose null hypothesis σ = 0.022 inch for certain wire rope diameter. Its alternate hypothesis is against σ ≠ 0.022 inch for a random sample of 18 yields i.e., S2 = 0.000324. take the level of significance as 0.05.

Solution:

• H0 : σ = 0.022
• H1 : σ ≠ 0.022
• Statistics:

X2 =

Here,

S2 = Sample Variance (Unbiased)

It follows the chi-square distribution with the (n – 1) degree of freedom.

• Here, α = 0.05 and the alternative hypothesis will be right tailed test

i.e. α/ 2 = 0.025

X20.025, 17= 30.191

• Computation

X2cal =

• Decision:

We have seen that

Xcal< 30.191

X2cal> 7.564

Thus, it will lie in the accepted region

Therefore,

H0 is accepted and true diameter of wire rope is 0.022 inch.  