In: Statistics and Probability
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
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 |