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The slope of a line in regression analysis tells us how much the dependent variable (often represented as \( y \)) is expected to change as the independent variable (\( x \)) increases by one unit. Calculating the slope is crucial because it helps us understand the relationship between \( x \) and \( y \).

To find the slope, \( \beta_1 \), we use the formula:\[ \beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]Where:

• \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) is the covariance between \( x \) and \( y \), which measures how much \( x \) and \( y \) change together.
• \( \sum (x_i - \bar{x})^2 \) is the variance of \( x \), showing how spread out the \( x \) values are.

In the provided exercise, after calculating these values with the given data, we find that the slope \( \beta_1 \) is \(-1.25\). This negative slope indicates that for every additional unit of \( x \), the expected value of \( y \) decreases by 1.25 units. Understanding the slope helps us see the direction and strength of the relationship between the two variables.

The y-intercept in the context of a regression equation represents the expected value of the dependent variable \( y \) when the independent variable \( x \) is 0. It's the point at which the regression line crosses the y-axis.

To calculate the y-intercept, \( \beta_0 \), use:\[ \beta_0 = \bar{y} - \beta_1 \cdot \bar{x} \]Where:

• \( \bar{y} \) is the mean of the y-values.
• \( \beta_1 \) is the slope of the regression line.
• \( \bar{x} \) is the mean of the x-values.

In our example, substituting the calculated means and the slope value, we find that \( \beta_0 \) equals 16.6875. This means when \( x \) is 0, the model predicts that \( y \) would be approximately 16.6875. The y-intercept helps us anchor the regression line on the graph, providing a starting point for mapping out the line.

The regression equation represents the relationship between our independent and dependent variables through a simple linear equation format. By connecting the slope and y-intercept, the equation predicts values of \( y \) for given values of \( x \).

The standard form of a regression equation is:\[ \hat{y} = \beta_0 + \beta_1 x \]Where:

• \( \hat{y} \) is the predicted value of the dependent variable.
• \( \beta_0 \) is the y-intercept.
• \( \beta_1 \) is the slope of the line.

For the given exercise, using the determined values of \( \beta_0 \) and \( \beta_1 \), the regression equation is \( \hat{y} = 16.6875 - 1.25x \). This equation can be used to predict \( y \) for any given \( x \), like when \( x = 7 \), resulting in \( \hat{y} = 8.9375 \). The regression equation is a powerful tool that provides insights into data trends and helps make informed predictions.