What is the Hypothesis Test for Regression Slope, also known as the t-test for the regression slope?

The Hypothesis Test for Regression Slope, also known as the t-test for the regression slope, is a statistical test used to determine whether the slope coefficient of a regression model is significantly different from zero. This test assesses whether there is a linear relationship between the independent variable (predictor) and the dependent variable (response) in a regression analysis.

Here's how the t-test for the regression slope typically works:

1. Null Hypothesis (H₀): The null hypothesis states that there is no linear relationship between the independent variable and the dependent variable. Specifically, it suggests that the slope coefficient β₁ in the regression model is equal to zero.

H₀: β₁ = 0

2. Alternative Hypothesis (H1): The alternative hypothesis contradicts the null hypothesis, suggesting that there is a significant linear relationship between the independent and dependent variables. Specifically, it implies that the slope coefficient β₁ in the regression model is not equal to zero.

H_1: β₁ ≠ 0

3. Test Statistic: The test statistic used for the t-test of the regression slope is calculated as:

t = (β₁ - 0) ⁄ SE(β₁)

Where:

- β₁ is the estimated slope coefficient from the regression model.

- SE(β₁) is the standard error of the slope coefficient.

4. Degrees of Freedom: The degrees of freedom for the t-test of the regression slope is typically n - 2 , where n is the sample size and 2 accounts for the two parameters estimated in the regression model (intercept and slope).

5. Decision Rule: The decision to reject or fail to reject the null hypothesis is based on the calculated t-statistic compared to the critical value from the t-distribution with n - 2 degrees of freedom at a chosen significance level (e.g., α = 0.05 ).

If the absolute value of the calculated t-statistic is greater than the critical value, then the null hypothesis is rejected, indicating that there is sufficient evidence to conclude that there is a significant linear relationship between the variables. Otherwise, if the absolute value of the calculated t-statistic is not greater than the critical value, the null hypothesis is not rejected, suggesting insufficient evidence to conclude a significant linear relationship.

6. Conclusion: Based on the decision rule, you conclude whether there is sufficient evidence to support the presence of a linear relationship between the variables or not.

The t-test for the regression slope is a fundamental tool in regression analysis, providing insights into the significance of the relationship between variables and the effectiveness of the regression model in explaining the variability in the dependent variable.

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