To determine the slope ($b_1$) and y-intercept ($b_0$) of the least-squares regression line, we use the following formulas:

1. Slope ($b_1$): $b_1 = r \cdot \frac{s_y}{s_x}$ Where:

• $r$ is the correlation coefficient
• $s_y$ is the standard deviation of the response variable (mpg)
• $s_x$ is the standard deviation of the explanatory variable (weight)

Given:

• $r = -0.959$
• $s_y = 9.15$
• $s_x = 6.02$

So, the slope is: $b_1 = -0.959 \cdot \frac{9.15}{6.02}$

2. Y-intercept ($b_0$): $b_0 = \bar{y} - b_1 \cdot \bar{x}$ Where:

• $\bar{y}$ is the mean of the response variable (mpg)
• $\bar{x}$ is the mean of the explanatory variable (weight)

Given:

• $\bar{y} = 19.8$
• $\bar{x} = 12.8$

Let's first calculate the slope and then use it to find the y-intercept.

Step 1: Calculate the slope

$b_1 = -0.959 \cdot \frac{9.15}{6.02}$

Let's calculate this value.The calculated slope is approximately $b_1 = -1.46$.

Step 2: Calculate the y-intercept

Using the formula for the y-intercept: $b_0 = \bar{y} - b_1 \cdot \bar{x}$ Substitute the values: $b_0 = 19.8 - (-1.46) \cdot 12.8$

Now, let's compute the y-intercept.The calculated y-intercept is approximately $b_0 = 38.49$.

Conclusion:

The correct slope and y-intercept for the least-squares regression line are:

• Slope = $-1.46$
• Y-intercept = $38.49$

Therefore, the correct answer is C.

Would you like further details or have any other questions? Here are some related questions for deeper understanding:

1. How is the correlation coefficient interpreted in the context of weight and miles per gallon?
2. What is the significance of a negative slope in a regression line?
3. How would changing the units of weight or mpg affect the regression equation?
4. Can the least-squares regression line be used for prediction? If so, how?
5. How does the value of the correlation coefficient affect the accuracy of the regression model?

Tip: Always check the signs of the slope and intercept to ensure they align with the relationship (positive or negative) observed in the data.