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Significance Test of Regression parameter (Z test and T test)

Significance Test of Regression parameter (Z test and T test)

05/03/2021
Significance Test of Regression parameter (Z test and T test)
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Significance Test of Regression parameter

In significance test, of the regression coefficient, we test whether the given regression coefficient is significant or not. In another word, these tests are performed to know the relation between the dependent and independent variables. There are different tests for regression coefficient which are discussed below

Z-TEST

When the sample size is large or n ≥ 30 then the Z test is performed otherwise student t-test is performed. It tests the relationship between the dependent and independent variables. If there is a relationship between the dependent and independent variables then the test is said to be significant otherwise the test is said to be non-significant.

Null hypothesis (H­0): β = 0

There is no relationship between the dependent and independent variables Y and X. or the test is not significant.

Alternative Hypothesis (H1): β ≠ 0 (two-tailed test)

There is a linear relationship between the dependent and independent variables.  Or test is significant.

Test stat­

Test stat for Z test is given by

Z=estimator-parameterS.E. of parameter

  

 

  

where S.E. (β̂) = σSxx

where σ is the standard deviation of Y, σ = S.E. =    =   

where K=2 is a number of variables and n is the sample size or the number of observations.

 

And Sxx is the corrected sum of the square of x  which is given by the

 

 

Critical value and level of significance

The value of Z at α level of significance can be calculated from the table. The level of signs indicates the relationship between the dependent and independent variables which is high or low. If the level of significance is not given then we use the 5% level of significance.

Decision

If the tabulated value of Z at α level of significance is greater than the calculated value I.e. Ztab > Zcal then H0 is accepted otherwise rejected.

T-TEST

When the sample size is small or n < 30 then a t-test is performed otherwise Z test is performed. It tests the relationship between the dependent and independent variables. If there is a relationship between the dependent and independent variables then the test is said to be significant otherwise the test is said to be non-significant.

Null hypothesis (H­0): β = 0

There is no relationship between the dependent and independent variables Y and X. or the test is not significant

Alternative Hypothesis (H1): β ≠ 0 (two-tailed test)

There is a linear relationship between the dependent and independent variables.  Or test is significant

Test stat­

Test stat for the t-test is given by

t=estimator-parameterS.E. of parameter

  

 

where S.E. (β̂) =  where σ̂ is an unbiased estimator of σ

where σ is the standard deviation of Y, σ̂ = S.E. =    =   and

where K=2 is a number of variables and n is the sample size or a number of observations. And Sxx is the corrected sum of the square of x which is given by the

 

Critical value and d.f.

The value of t at  level of significance and at the n-2 degree of freedom can be calculated from the table. The level of signs indicates the relationship between the dependent and independent variables which is high or low. If a level of significance is not given then we use the 5% level of significance

Decision

If the tabulated value of t at  level of significance and at n-2 d.f. is greater than the calculated value I.e. ttab > tcal then H0 is accepted otherwise rejected. When H0 is rejected the alternative Hypothesis H1 is accepted

Z- Table 

 

Z- Table-1

Nature Level of significance
 value of  α 1% 2% 3% 4% 5%
two tail  2.576 2.326 2.054 1.960 1.645
One tail 2.326 2.054 1.751 1.645 1.282

 

 

 

 

 

 

table-2

Z- Table-2
Nature Level of significance

confidence level (1- α) 

50% 68.2% 90% 95% 96% 98% 99% 99.73%
α 50% 72.72% 10% 5% 4% 2% 1% 0.27%
Zα   0.6745 1 1.645 1.96 2.05 2.33 2.58 3

 

 

 


 

 

 

How to Use Z table and T table (In Details)
1. Calculate the Value of Z or T using the formula given above. 
2. For example, You got the calculated value of the Z test at α = 5% is  2.30. Hence Zcal =2.30.
3. Now, look at the table at α = 5%, value of Z is 1.960. (from z-table-2)
4. Here value Ztab=2.30 >Zcal = 1.960 . Hence null hypothesis is accepted.  (otherwise rejected)

 

 

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