Question

Let x be the age in years of a licensed automobile driver. Let y be the percentage of all fatal accidents (for a given age) due to speeding. For example, the first data pair indicates that 36% of all fatal accidents of 17-year-olds are due to speeding. x 17 27 37 47 57 67 77 y 36 25 23 12 10 7 5 Complete parts (a) through (e), given Σx = 329, Σy = 118, Σx2 = 18,263, Σy2 = 2768, Σxy = 4126, and r ≈ −0.962. (a) Draw a scatter diagram displaying the data.

(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

(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) Predict the percentage of all fatal accidents due to speeding for 25-year-olds. (Round your answer to two decimal places.)
%

Answer 1

a) Scatter diagram of the data is given by,

b)

X	Y	X^2	Y^2	XY
17	36	289	1296	612
27	25	729	625	675
37	23	1369	529	851
47	12	2209	144	564
57	10	3249	100	570
67	7	4489	49	469
77	5	5929	25	385
329	118	18263	2768	4126

Σx	=	329
Σy	=	118
Σx2	=	18263
Σy2	=	2768
Σxy	=	4126

correlation coefficient,

r = - 0.962

c) the equation of the least-squares line  = a + bx.

  

b = - 0.507

a = 40.686

The least square regression equation is y = 40.686 - 0.507 x

d) The coefficient of determination is given by,

r2 = (-0.962)2 = 0.925

The percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line is 92.5% .

The percentage of the variation in y can be unexplained by the corresponding variation in x and the least-squares line is (100-92.5) = 7.5%

e) The percentage of all fatal accidents due to speeding for 25-year-olds is,

y = 40.686 - 0.507*25 = 28.01%

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