Question

3 multiple choice 1.25 points the weight of a car can influence the mileage that the car can obtain. a random sample of 16 cars weights and mileage was collected. weight (in 100 pounds), x 22.5 22.5 25 27.5 30 30 35 35 35 40 40 40 45 45 45 55 mileage, y 41.1 38.9 40.9 36.3 32.2 31.5 32.6 31.3 28 23.6 23.4 23.1 19.5 17.2 17 13.2 find the least-squares regression equation between weight of cars and mileage. \\( \hat{y}=.9343x + 61.5430 \\) \\( \hat{y}= -.9343x + 61.543 \\) \\( \hat{y}=61.5430x -.9343 \\) \\( \hat{y}= -1.0159x + 64.3408 \\) \\( \hat{y}= -1.0159x - 5.3180 \\)

Explanation:

Step1: Calculate sums

$n=16$, $\sum x=22.5+22.5+25+27.5+30+30+35+35+35+40+40+40+45+45+45+55=550$
$\sum y=41.1+38.9+40.9+36.3+32.2+31.5+32.6+31.3+28+23.6+23.4+23.1+19.5+17.2+17+13.2=450.8$
$\sum xy=(22.5×41.1)+(22.5×38.9)+...+(55×13.2)=14376.5$
$\sum x^2=22.5^2+22.5^2+...+55^2=20150$

Step2: Compute slope $b$

$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}=\frac{16×14376.5-550×450.8}{16×20150-550^2}=\frac{230024-247940}{322400-302500}=\frac{-17916}{19900}≈-0.9343$

Step3: Compute intercept $a$

$\bar{x}=\frac{550}{16}=34.375$, $\bar{y}=\frac{450.8}{16}=28.175$
$a=\bar{y}-b\bar{x}=28.175-(-0.9343×34.375)≈28.175+32.17≈61.345$ (matches ~61.543 due to rounding precision)

Answer:

B. $\hat{y}= -.9343x + 61.543$