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 37% of all fatal accidents of 17-year-olds are due to speeding.

x 17,27,37,47,57,67,77

y 37,25,18,12,10,7,5

Complete parts (a) through (e), given Σx = 329, Σy = 114, Σx² = 18,263, Σy² = 2636, Σxy = 3958, and r ≈ −0.948.

(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=

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) Predict the percentage of all fatal accidents due to speeding for 45-year-olds. (Round your answer to two decimal places.)

=%

Answer 1

a.

b.

	x	y	x^2	y^2	xy
	17	37	289	1369	629
	27	25	729	625	675
	37	18	1369	324	666
	47	12	2209	144	564
	57	10	3249	100	570
	67	7	4489	49	469
	77	5	5929	25	385
Total	329	114	18263	2636	3958

X Values
∑ = 329
Mean = 47
∑(X - M_x)² = SS_x = 2800

Y Values
∑ = 114
Mean = 16.286
∑(Y - M_y)² = SS_y = 779.429

X and Y Combined
N = 7
∑(X - M_x)(Y - M_y) = -1400

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -1400 / √((2800)(779.429)) = -0.948

c.

Sum of X = 329
Sum of Y = 114
Mean X = 47
Mean Y = 16.2857
Sum of squares (SS_X) = 2800
Sum of products (SP) 6= -1400

Regression Equation = ŷ = bX + a

b = SP/SS_X = -1400/2800 = -0.5

a = M_Y - bM_X = 16.29 - (-0.5*47) = 39.786

ŷ = -0.5X + 39.786

d.

e. Here r= -0.948, so r^2=0.899

So explained variation is 89.8%

Unexplained variation is 10.2%

f. For x=45,

ŷ = (-0.5*45)+ 39.786=-5.21