Varience ratio test

VarienceVarience Ratio Test (test of significance difference between two population Variance).

To test whether the two independent estimates of population variances are significantly different or not, or are uniform or not to test whether the two independent estimates of the samples have come from the same universe and have a common variance, we use variance ratio set. Suppose that n₁,n₂ are samples drawn from a normal population with corresponding means µ₁, µ₂ and variance (σ1²,) (σ2²). Then the following steps are followed to test the variance ratio test.

Step I: Null hypothesis. H0:σ₁²,=σ₂²

That is no significant difference between two population variance or there is no significant difference between two the independent estimates of the common population variance or two population variance are same.

Step 2. Alternative hypothesis.H₁≠σ₂²=σ²

That is there is significance difference between two population variances, or there is the significant difference between two independent estimates of the common population variance, or two population variance are not same.

Step:3 Test statistic

Under H₀, the statistic is.

$$F=F=\frac{S_1^2}{S_2^2}∼F(V_1.V_2)IfS_1^2>S_2^2$$

$$and\,F=\frac{S_2^2}{S_1^2}∼F(V_2.V_1)IfS_2^2>S_1^2$$

$$Where\,S_1^2=Unbiased\,estimate\,of\,first\,population\,Varience$$

$$Where\,S_2^2=Unbiased\,estimate\,of\,second\,population\,Varience$$

If raw data is given, in this situation, we use following formula to calculate unbiased estimate of population variences.

$$S_1^2=\frac{1}{n_1-1}\sum(X_1-\overline{X}_1)^2[Actual\,mean\,Method]$$

$$=\frac{1}{n_1-1}[\sum(X_1^2-\frac{\sum(X_1)^2}{n_1}]\,[Direct\,method]$$

$$=\frac{1}{n_1-1}[\sum(d_1^2-\frac{\sum(d_1)^2}{n_1}]\,[Shortcut\,method]$$

$$S_2^2=\frac{1}{n_2-1}\sum(X_2-\overline{X}_2)^2[Actual\,mean\,Method]$$

$$=\frac{1}{n_2-1}[\sum(X_2^2-\frac{\sum(X_2)^2}{n_2}]\,[Direct\,method]$$

$$=\frac{1}{n_2-1}[\sum(d_2^2-\frac{\sum(d_2)^2}{n_2}]\,[Shortcut\,method]$$

$$Where\,d_1=X_1-A_1\,,A_1=assumed\,mean\,taken\,from\,X_1$$

$$Where\,d_2=X_2-A_1\,,A_2=assumed\,mean\,taken\,from\,X_2$$

If biased results I,e s₁ and s₂ or (s12 and s22) are given, in this situation, we use the following the formula to calculate unbiased estimates of population variance

$$S_1^2=\frac{n_1S_1^2}{(n_1-1)}$$

$$S_2^2=\frac{n_2S_2^2}{(n_2-1)}$$

$$Here\,,S_1^2=sample\,varience\,of\,first\,population\,variable$$

$$S_1^2=\frac{1}{n_1}\sum(X_1-\overline{X}_1)^2[Actual\,mean\,Method]$$

$$=\frac{1}{n_1}[\sum(X_1^2-\frac{\sum(X_1)^2}{n_1}]\,[Direct\,method]$$

$$=[\sum(d_1^2-\frac{\sum(d_1)^2}{n_1}]\,[Shortcut\,method]$$

$$S_2^2=\frac{1}{n_2}\sum(X_2-\overline{X}_2)^2[Actual\,mean\,Method]$$

$$=\frac{1}{n_2}[\sum(X_2^2-\frac{\sum(X_2)^2}{n_2}]\,[Direct\,method]$$

$$=\frac{1}{n_2}[\sum(d_2^2-\frac{\sum(d_2)^2}{n_2}]\,[Shortcut\,method]$$

Step:4 Level of significance.

We have to set 1% or 5% level os significance.(α)

Step:5 Degree of freedom:

$$(n_1-1,n_2-1)\,If\,F=\frac{S_1^2}{S_2^2}$$

$$or\,n_2-1,n_1-1),If\,F=\frac{S_1^2}{S_2^2}$$

Step 6. Critical value.

We have to determine the tabulated value of F at (α)% level of significance for required degree of freedom from F table.

Step 7. Decision.

If the calculated value of F(≤) tabulated value of F, we accept null hypothesis otherwise, we reject null hypothesis I,e accept the alternative hypothesis.

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