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	2	2	12	16	26	36
y	45	53	74	100	150	200

Complete parts (a) through (e), given Σx = 94, Σy = 622, Σx² = 2380, Σy² = 82,810, Σxy = 13,784, and r ≈ 0.990.

(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

(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.)
kg

Answer 1

a)

b)

X	Y	XY	X²	Y²
2	45	90	4	2025
2	53	106	4	2809
12	74	888	144	5476
16	100	1600	256	10000
26	150	3900	676	22500
36	200	7200	1296	40000

	X	Y	XY	X²	Y²
total sum	94	622	13784	2380	82810

SSxx =	Σx² - (Σx)²/n =	907.33
SSxy=	Σxy - (Σx*Σy)/n =	4039.33
SSyy =	Σy²-(Σy)²/n =	18329.33

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.990

c)

here, x̅ =Σx/n =   15.67   ,   ȳ = Σy/n =   103.67
estimated slope , ß1 = SSxy/SSxx =   4039.333   /   907.3333   =   4.4519
                  
intercept,   ß0 = y̅-ß1* x̄ =   33.9206          
                  
so, regression line is   Ŷ =   33.921 +   4.452 *x

d)

e)

R² =    (SSxy)²/(SSx.SSy) =    0.9811
explained = 98.1 %

unexplained = 1.9%

f)

Predicted Y at X=   24   is          
Ŷ=   33.92065   +   4.45187   *24=   140.77