Question

In: Statistics and Probability

Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x...

Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x be the weight of the car (in hundreds of pounds), and let y be the miles per gallon (mpg).

x 25 42 33 47 23 40 34 52
y 31 22 25 13 29 17 21 14

Complete parts (b) through (e), given Σx = 296, Σy = 172, Σx2 = 11,676, Σy2 = 4006, Σxy = 5924, and

r ≈ −0.932.


(b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

Σx =   
Σy =
Σx2 =
Σy2 =
Σxy =
r =


(c) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x =
y =
= +   x


(d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.



(e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained     %
unexplained     %


(f) Suppose a car weighs x = 42 (hundred pounds). What does the least-squares line forecast for y = miles per gallon? (Round your answer to two decimal places.)
mpg

Solutions

Expert Solution

b)

ΣX = 296.000
ΣY= 172.000
ΣX2 = 11676.000
ΣY2 = 4006.000
ΣXY = 5924.000
r = -0.932

c)

X̅=ΣX/n = 37.00
Y̅=ΣY/n = 21.50
ŷ = 43.986+(-0.608)x

e)

coefficient of determination r2 = 0.868
explained = 86.8%
unexplained= 13.2%

f)

predicted value =43.986+-0.608*42= 18.45

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