Question

You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weight of the calf (in kilograms).

x	1	2	11	16	26	36
y	46	52	79	100	150	200

Complete parts (a) through (e), given Σx = 92, Σy = 627, Σx² = 2354, Σy² = 83,561, Σxy = 13,719, and r ≈ 0.995.

(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 =	2
Σy =	3
Σx2 =	4
Σy2 =	5
Σxy =	6
r =	7

(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) The calves you want to buy are 24 weeks old. What does the least-squares line predict for a healthy weight? (Round your answer to two decimal places.)

Answer 1

Part a)

part b)

ΣX = 92
ΣY = 627
ΣX * Y = 13719
ΣX2 = 2354
ΣY2 = 83561

r = 0.995

Part c)

X̅ = Σ( Xi / n ) = 92/6 = 15.33
Y̅ = Σ( Yi / n ) = 627/6 = 104.5

Equation of regression line is Ŷ = a + bX


b = 4.352
a =( Σ Y - ( b * Σ X) ) / n
a =( 627 - ( 4.3516 * 92 ) ) / 6
a = 37.776
Equation of regression line becomes Ŷ = 37.776 + 4.352 X

part e)

Coefficient of Determination
= 0.990
Explained variation = 0.99* 100 = 99.0%
Unexplained variation = 1 - 0.99* 100 = 1.0%

part f)
When X = 24
Ŷ = 37.776 + 4.352 X
Ŷ = 37.776 + ( 4.352 * 24 )
Ŷ = 142.22