Linear Regression | Hypothesis Testing

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I saw this screen in a video for t-test for slope. Can you shed some light as to what is being taught here?

Answer

The screen you are seeing explains how to perform a t-test for the slope of a simple linear regression model. The goal is to assess whether the slope ($b_1$) of the regression line is statistically significantly different from zero, which would indicate a linear relationship between the dependent and independent variables.
Here's an explanation of the components shown in the image:
t-Statistic for the Slope:
Formula: $t = \frac{b_1}{SE(b_1)}$
$b_1$ is the estimated slope of the regression line.
$SE(b_1)$ is the standard error of the slope estimate, which measures the variability of the slope coefficient across different samples.
This formula is used to calculate the t-statistic for the slope. The t-statistic quantifies how many standard errors the estimated slope is away from zero.
Standard Error of the Slope $SE(b_1)$:
The standard error $SE(b_1)$ of the slope is calculated using the formula:
$$SE(b_1) = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}} \div \sqrt{\sum (x_i - \bar{x})^2}$$
Numerator:
$\sum (y_i - \hat{y}_i)^2$ is the sum of squared residuals (SSR), where $y_i$ are the actual values of the dependent variable and $\hat{y}_i$ are the predicted values from the regression model.
$n - 2$ accounts for the degrees of freedom. Since two parameters are estimated in a simple linear regression (the intercept and the slope), we subtract 2 from the total number of data points $n$.
Denominator:
$\sum (x_i - \bar{x})^2$ is the sum of squared deviations of the independent variable $x$ from its mean $\bar{x}$. This reflects how much variability exists in the independent variable $x$.
Interpretation:
t-Statistic: Once the t-statistic is calculated, you compare it to the critical value from the t-distribution (based on the degrees of freedom and the desired significance level, e.g., 0.05). If the t-statistic is greater in magnitude than the critical value, you reject the null hypothesis that the slope is zero.
Null Hypothesis ($H_0$): $b_1 = 0$. This means that there is no linear relationship between the variables.
Alternative Hypothesis ($H_1$): $b_1 \neq 0$. This suggests a linear relationship between the variables.
If the null hypothesis is rejected, it implies that there is a statistically significant linear relationship between the independent and dependent variables.
In summary, this screen is demonstrating how to perform a t-test on the slope of a regression line to check whether the slope is significantly different from zero, indicating a linear relationship between the two variables. The t-test is based on the ratio of the slope to its standard error, and the standard error itself is calculated using the residuals from the regression and the variability of the independent variable.

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