Which of the following linear regression equations best represents the fuel efficiency?

Steps to Solve

1. Identify Variables Let ( x ) be the speed (in miles per hour) and ( y ) be the fuel efficiency (in miles per gallon) based on the data provided:

2. Calculate the Slope ( m ) The slope ( m ) can be calculated using the formula:

$$ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} $$

Where ( N ) is the number of data points. In our case, ( N = 4 ).

First, we need to calculate:

• ( \sum x = 40 + 50 + 60 + 70 = 220 )
• ( \sum y = 80 + 65 + 60 + 55 = 260 )
• ( \sum xy = (40 \cdot 80) + (50 \cdot 65) + (60 \cdot 60) + (70 \cdot 55) = 3200 + 3250 + 3600 + 3850 = 13900 )
• ( \sum x^2 = 40^2 + 50^2 + 60^2 + 70^2 = 1600 + 2500 + 3600 + 4900 = 12600 )

Now substitute back:

$$ m = \frac{4(13900) - (220)(260)}{4(12600) - (220)^2} $$

3. Calculate the Intercept ( b ) The intercept ( b ) is calculated using the formula:

$$ b = \frac{\sum y - m(\sum x)}{N} $$

4. Construct the Linear Regression Equation Using the calculated values of ( m ) and ( b ), we can form the equation:

$$ y = mx + b $$

5. Compare with Given Options Convert the equation into a standard format and check which of the provided equations matches:

• A: $y = 8x + 109$
• B: $y = .8x - 109$
• C: $y = -0.8x + 109$
• D: $y = 2x + 54$