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 5 11 16 26 36 y 39 47 73 100 150 200 Complete parts (a) through (e), given Σx = 95, Σy = 609, Σx2 = 2375, Σy2 = 81,559, Σx y = 13,777, and r ≈ 0.997.

Verify the given sums Σx, Σy, Σx², Σy², Σx y, and the value of the sample correlation coefficient r. (For each answer, enter a number. Round your value for r to three decimal places.)

Σx =

Σy =

Σx² =

Σy² =

Σx y =

r =

(c)

Find , and . Then find the equation of the least-squares line  = a + b x. (For each answer, enter a number. Round your answers for  and  to two decimal places. Round your answers for a and b to three decimal places.)

= x bar =

= y bar =

= value of a coefficient

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? (For each answer, enter a number. 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 20 weeks old. What does the least-squares line predict for a healthy weight (in kg)? (Enter a number. Round your answer to two decimal places.)
kg

Answer 1

b.

X Values
∑ = 95
Mean = 15.833
∑(X - Mx)2 = SSx = 870.833

Y Values
∑ = 609
Mean = 101.5
∑(Y - My)2 = SSy = 19745.5

X and Y Combined
N = 6
∑(X - Mx)(Y - My) = 4134.5

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = 4134.5 / √((870.833)(19745.5)) = 0.997

c.

Sum of X = 95
Sum of Y = 609
Mean X = 15.8333
Mean Y = 101.5
Sum of squares (SSX) = 870.8333
Sum of products (SP) = 4134.5

Regression Equation = ŷ = bX + a

b = SP/SSX = 4134.5/870.83 = 4.748

a = MY - bMX = 101.5 - (4.75*15.83) = 26.327

ŷ = 4.748X + 26.327

e. Here r=0.997

So r^2=0.997^2=0.994

So explained variation is 99.4%

Unexplained variation is 0.6%

f. For x=20, ŷ = 4.748*20+ 26.327=121.29