Solution

The given regression equation is: $\hat{y} = 17.6 + 3.8x_1 - 2.3x_2 + 7.6x_3 + 2.7x_4$

Part a: Interpret $b_1$, $b_2$, $b_3$, and $b_4$

1. $b_1 = 3.8$: For every 1-unit increase in $x_1$, $y$ increases by 3.8, holding $x_2$, $x_3$, and $x_4$ constant.
2. $b_2 = -2.3$: For every 1-unit increase in $x_2$, $y$ decreases by 2.3, holding $x_1$, $x_3$, and $x_4$ constant.
3. $b_3 = 7.6$: For every 1-unit increase in $x_3$, $y$ increases by 7.6, holding $x_1$, $x_2$, and $x_4$ constant.
4. $b_4 = 2.7$: For every 1-unit increase in $x_4$, $y$ increases by 2.7, holding $x_1$, $x_2$, and $x_3$ constant.

Part b: Predict $y$ when $x_1 = 10$, $x_2 = 5$, $x_3 = 1$, and $x_4 = 2$

Substitute the values into the equation:

$\hat{y} = 17.6 + 3.8(10) - 2.3(5) + 7.6(1) + 2.7(2)$

Simplify step by step:

1. $17.6 + 3.8(10) = 17.6 + 38 = 55.6$
2. $55.6 - 2.3(5) = 55.6 - 11.5 = 44.1$
3. $44.1 + 7.6(1) = 44.1 + 7.6 = 51.7$
4. $51.7 + 2.7(2) = 51.7 + 5.4 = 57.1$

$\hat{y} = 57.1$

Final Answers:

a.

• $b_1 = 3.8$
• $b_2 = -2.3$
• $b_3 = 7.6$
• $b_4 = 2.7$

b. Predicted $y = 57.1$

Would you like further clarification on this or additional steps? Here are five related questions to explore:

1. How does multicollinearity affect interpretation of regression coefficients?
2. What does the intercept $17.6$ represent in this context?
3. How would predicted $y$ change if $x_2$ doubled in value while others stayed constant?
4. What is the significance of a negative regression coefficient like $b_2$?
5. How could we evaluate the goodness-of-fit of this regression model?

Tip: When solving regression problems, always ensure that coefficients are interpreted with respect to other variables held constant.