To analyze the relationship between the weight of vehicles and their gas mileage, let's determine the equation of the least-squares regression line, where [tex]\( y \)[/tex] is the predicted gas mileage and [tex]\( x \)[/tex] is the weight (in tons) of the vehicle.
Given the data:
| Weight (tons) | Gas mileage (mpg) |
| ------------- | ----------------- |
| 1.6 | 29 |
| 1.6 | 45 |
| 1.75 | 26 |
| 1.95 | 22 |
| 2 | 18 |
| 2 | 21 |
| 2.3 | 21 |
| 2.5 | 18 |
We perform a least-squares linear regression to find the equation of the regression line in the form:
[tex]\[ \hat{y} = mx + b \][/tex]
Here, [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the intercept. Through the regression analysis, we find:
[tex]\[ m = -20.175131348511385 \][/tex]
[tex]\[ b = 64.59369527145357 \][/tex]
Thus, the equation of the regression line is:
[tex]\[ \hat{y} = -20.175131348511385x + 64.59369527145357 \][/tex]
Now we use this equation to predict the gas mileage for a car that weighs 1.8 tons. Plugging [tex]\( x = 1.8 \)[/tex] into the regression equation:
[tex]\[ \hat{y} = -20.175131348511385 \cdot 1.8 + 64.59369527145357 \][/tex]
[tex]\[ \hat{y} \approx 28.278458844133077 \][/tex]
Therefore, according to the regression equation, a car that weighs 1.8 tons would have a gas mileage of approximately 28.28 miles per gallon.
