91Ó°ÊÓ

In regression analysis, coefficients are crucial components of the regression equation. They tell us how much the dependent variable, often denoted as \(y\), is expected to increase or decrease with a one-unit change in one of the independent variables, while keeping all other variables constant.

For the given regression equation: \[\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}\] we have four coefficients, each corresponding to one of the independent variables \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\).

• \(b_1 = 3.8\): For each additional unit of \(x_1\), the predicted \(y\) increases by 3.8 units.
• \(b_2 = -2.3\): For each additional unit of \(x_2\), the predicted \(y\) decreases by 2.3 units, indicating an inverse relationship.
• \(b_3 = 7.6\): For each additional unit of \(x_3\), the predicted \(y\) increases by 7.6 units.
• \(b_4 = 2.7\): For each additional unit of \(x_4\), the predicted \(y\) increases by 2.7 units.

Interpreting coefficients in a regression equation helps us understand the impact of each variable on the predicted outcome. Let's explore each coefficient in our regression context.

The **coefficient \(b_1 = 3.8\)** suggests that, with an increase in \(x_1\) by one unit, the dependent variable \(y\) is expected to rise by 3.8 units. All other variables remain unchanged during this interpretation.

The **coefficient \(b_2 = -2.3\)** indicates a decrease of 2.3 units in \(y\) for each unit increment in \(x_2\). This negative coefficient shows an inverse relationship between \(x_2\) and \(y\).

When considering **\(b_3 = 7.6\)**, a one-unit increase in \(x_3\) leads to a rise in \(y\) by 7.6 units. The correlation here is positive, suggesting that as \(x_3\) increases, so does \(y\).

Finally, **\(b_4 = 2.7\)** implies that \(y\) will increase by 2.7 units with each one-unit rise in \(x_4\). The positive sign reflects a direct positive influence of \(x_4\) on \(y\).

Predicted values in regression help determine what \(y\) would be, based on specific values of the independent variables, \(x\). Let's use the provided regression equation and plug in the values for \(x_1 = 10\), \(x_2 = 5\), \(x_3 = 1\), and \(x_4 = 2\).

To find the predicted value of \(y\), substitute these values:
\[ y = 17.6 + 3.8(10) - 2.3(5) + 7.6(1) + 2.7(2) \]
Calculate each component separately:

• \(3.8 \times 10 = 38.0\)
• \(-2.3 \times 5 = -11.5\)
• \(7.6 \times 1 = 7.6\)
• \(2.7 \times 2 = 5.4\)

Add these results to the intercept, 17.6, to find the predicted \(y\):
\[ y = 17.6 + 38.0 - 11.5 + 7.6 + 5.4 = 57.1 \]

Thus, the estimated value of \(y\) given the specified \(x\) values is 57.1. This process reveals how the regression model predicts \(y\) using specific input values.