Question

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 17 33 42 Complete parts (a) through (e), given Σx = 372, Σy = 115, Σx2 = 24814, Σy2 = 3331, Σxy = 8465, and r ≈ 0.951. (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 y hat = 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 hat = + 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 failing to yield the right of way for 70-year-olds. (Round your answer to two decimal places.) %

Answer 1

Part a)

Part b)

	X	Y	X * Y	X2	Y2
	37	5	185	1369	25
	47	8	376	2209	64
	57	10	570	3249	100
	67	17	1139	4489	289
	77	33	2541	5929	1089
	87	42	3654	7569	1764
Total	372	115	8465	24814	3331

r = 0.951

Part c)
X̅ = Σ( Xi / n ) = 372/6 = 62
Y̅ = Σ( Yi / n ) = 115/6 = 19.17

Equation of regression line is Ŷ = a + bX

b = ( 6 * 8465 - 372 * 115 ) / ( 6 * 24814 - ( 372 )2)
b = 0.763
a =( Σ Y - ( b * Σ X) ) / n
a =( 115 - ( 0.7629 * 372 ) ) / 6
a = -28.13
Equation of regression line becomes Ŷ = -28.130 + 0.763 X

Part e)
Coefficient of Determination
R2 = r2 = 0.904
Explained variation = 0.904* 100 = 90.4%
Unexplained variation = 1 - 0.904* 100 = 9.6%

Part f)
When X = 70
Ŷ = -28.13 + 0.763 X
Ŷ = -28.13 + ( 0.763 * 70 )
Ŷ = 25.28