Questions: (a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. ŷ = -0.000640 x + 411 (Round the x coefficient to five decimal places as needed. Round the constant to one decimal place as needed) (b) Interpret the slope and y-intercept, if appropriate. Choose the correct answer below and fill in any answer boxes in your choice. (Use the answer from part a to find this answer) A. For every pound added to the weight of the car, gas mileage in the city will decrease by mile(s) per gallon, on average. A weightless car will get miles per gallon, on average. B. A weightless car will get miles per gallon, on average. It is not appropriate to interpret the slope. C. For every pound added to the weight of the car, gas mileage in the city will decrease by mile(s) per gallon, on average. It is not appropriate to interpret the y-intercept. D. It is not appropriate to interpret the slope or the y-intercept.

(a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. ŷ = -0.000640 x + 411 (Round the x coefficient to five decimal places as needed. Round the constant to one decimal place as needed) (b) Interpret the slope and y-intercept, if appropriate. Choose the correct answer below and fill in any answer boxes in your choice. (Use the answer from part a to find this answer) A. For every pound added to the weight of the car, gas mileage in the city will decrease by mile(s) per gallon, on average. A weightless car will get miles per gallon, on average. B. A weightless car will get miles per gallon, on average. It is not appropriate to interpret the slope. C. For every pound added to the weight of the car, gas mileage in the city will decrease by mile(s) per gallon, on average. It is not appropriate to interpret the y-intercept. D. It is not appropriate to interpret the slope or the y-intercept.
Transcript text: (a) Find the least-squares regession line treating weight as the explanatory variable and miles per gallon as the response variable. \[ \hat{y}=-000640 x+(411) \] (Round the $x$ coefficient to five decimal places as needed. Round the constant to one decimal place as needed) (b) Interpret the slope and $y$-intercept, if appropnate Choose the correct answer below and fill in any answer boxes in your choice. (Use the answer from part a to find this answer) A. For every pound added to the weight of the car, gas mileage in the city will decrease by $\square$ mile(s) per gallon, on average. A weightless car will get $\square$ miles per gallon, on average. B. A weightless car will get $\square$ miles per gallon, on average. It is not appropnate to interpret the slope. C. For every pound arded to the weight of the car, gas mileage in the city will decrease by $\square$ mile(s) per galion, on average. It is not appropriate to interpret the $y$-intercept. D. It is not appropnate to interpret the slope or the $y$-intercept.
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Solution

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Solution Steps

Step 1: Calculate Means

The means of the variables \( x \) (weight) and \( y \) (miles per gallon) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 2500.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 24.9 \]

Step 2: Calculate Correlation Coefficient

The correlation coefficient \( r \) is found to be:

\[ r = -0.99878 \]

Step 3: Calculate Slope (β)

The numerator for the slope \( \beta \) is calculated as:

\[ \text{Numerator for } \beta = \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 588100 - 10 \times 2500.0 \times 24.9 = -34400.0 \]

The denominator for the slope \( \beta \) is:

\[ \text{Denominator for } \beta = \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 65900000 - 10 \times 2500.0^2 = 3400000.0 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{-34400.0}{3400000.0} = -0.01012 \]

Step 4: Calculate Intercept (α)

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 24.9 - (-0.01012) \times 2500.0 = 50.19412 \]

Step 5: Formulate the Regression Equation

The equation of the least-squares regression line is:

\[ \hat{y} = 50.19412 - 0.01012x \]

Step 6: Interpret the Slope and Intercept

The interpretation of the slope and intercept is as follows:

  • For every pound added to the weight of the car, gas mileage in the city will decrease by \( 0.01012 \) mile(s) per gallon, on average.
  • A weightless car will get \( 50.19412 \) miles per gallon, on average.

Final Answer

The slope and intercept can be summarized as follows:

  • Slope (β): \( -0.01012 \)
  • Intercept (α): \( 50.19412 \)

Thus, the final answer is:

\[ \boxed{\text{Slope: } -0.01012, \text{ Intercept: } 50.19412} \]

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