Let x be the age of a licensed driver in years. Let y be the percentage of all fatal accidents (for a given age) due to failure to yield the right of way. For example, the first data pair states that 5% of all fatal accidents of 37-year-olds are due to failure to yield the right of way.
x | 37 | 47 | 57 | 67 | 77 | 87 |
y | 5 | 8 | 10 | 15 | 31 | 44 |
Complete parts (a) through (e), given Σx = 372, Σy = 113, Σx^{2} = 24814, Σy^{2} = 3311, Σxy = 8351, and r ≈ 0.935.
(a) Draw a scatter diagram displaying the data.
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(b) Verify the given sums Σx, Σy,
Σx^{2}, Σy^{2}, Σxy, and
the value of the sample correlation coefficient r. (Round
your value for r to three decimal places.)
Σx = | |
Σy = | |
Σx^{2} = | |
Σy^{2} = | |
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
r^{2}. 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 r^{2}
to three decimal places. Round your answers for the percentages to
one decimal place.)
r^{2} = | |
explained | % |
unexplained | % |
(f) Predict the percentage of all fatal accidents due to failing to
yield the right of way for 70-year-olds. (Round your answer to two
decimal places.)
%
Part b)
ΣX = 372
ΣY = 113
ΣX * Y = 8351
ΣX2 = 24814
ΣY2 = 3311
r = 0.935
Part c)
Equation of regression line is Ŷ = a + bX
b = 0.769
a =( Σ Y - ( b * Σ X) ) / n
a =( 113 - ( 0.7686 * 372 ) ) / 6
a = -28.818
Equation of regression line becomes Ŷ = -28.818 + 0.769
X
Part e)
Coefficient of Determination
= 0.874
Explained variation = 0.874* 100 = 87.4%
Unexplained variation = 1 - 0.874* 100 = 12.6%
Part f)
When X = 70
Ŷ = -28.818 + 0.769 X
Ŷ = -28.818 + ( 0.769 * 70 )
Ŷ = 25.01
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