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Consider normal population, N (µ, σ2)
- Set up H0 : σ2
- Set up H1 : σ2<σ02 or σ2>σ02, 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<X2tabi.e., X21 – α, n -1 |
| σ2> σ02 | X2cal> X2tab i.e., X2α, n -1 |
| σ2 ≠ σ02 | X2cal< X2tab i.e., X21 – α/ 2, n -1
or,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.
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
Links of Next Statistics Topics:-
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory
