# 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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