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

**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** (H_{0}): β = 0

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

**Alternative Hypothesis** (H_{1}): 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*-*parameter**S*.*E*. *of* *parameter*

where S.E. (β̂) = *σ**S**xx*

**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 S_{xx} 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. Z_{tab} > Z_{cal }then H_{0} 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** (H_{0}): β = 0

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

**Alternative Hypothesis** (H_{1}): 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*-*parameter**S*.*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 S_{xx} 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. t_{tab} > t_{cal }then H_{0} is accepted otherwise rejected. When H_{0} is rejected the alternative Hypothesis H_{1 }is accepted

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**

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% |

Z_{α } |
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 Z**Z

_{tab}=2.30 >_{cal }

**= 1.960 . Hence null hypothesis is accepted. (otherwise rejected)**

**Want To study More topics ?**

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