Question

1. 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.

(a) Make a scatter diagram of the data. (Select the correct graph.)

(b) 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 value of b coefficient

(d)

Graph the least-squares line. Be sure to plot the point (, ) as a point on the line. (Select the correct graph.)

(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? (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 11 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

X	Y	XY	X²	Y²
1	39	39	1	1521
5	47	235	25	2209
11	73	803	121	5329
16	100	1600	256	10000
26	150	3900	676	22500
36	200	7200	1296	40000

Ʃx =	95
Ʃy =	609
Ʃxy =	13777
Ʃx² =	2375
Ʃy² =	81559
Sample size, n =	6
x̅ = Ʃx/n = 95/6 =	15.83333333
y̅ = Ʃy/n = 609/6 =	101.5
SSxx = Ʃx² - (Ʃx)²/n = 2375 - (95)²/6 =	870.8333333
SSyy = Ʃy² - (Ʃy)²/n = 81559 - (609)²/6 =	19745.5
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 13777 - (95)(609)/6 =	4134.5

a) Scatter plot:

b)

Ʃx =    95

Ʃy =    609

Ʃx² =    2375

Ʃy² =    81559

Ʃxy =    13777

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 4134.5/√(870.83333*19745.5) =    0.997

c)

x̅ = Ʃx/n = 95/6 =    15.8333

y̅ = Ʃy/n = 609/6 =    101.5

Slope, b = SSxy/SSxx = 4134.5/870.83333 =    4.747751

y-intercept, a = y̅ -b* x̅ = 101.5 - (4.74775)*15.83333 =    26.32727

Regression equation :   

ŷ = 26.327 + (4.748) x  

d)

e)

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (4134.5)²/(870.83333*19745.5) = 0.994

Explained =    99.4%

Unexplained =    0.6%

f)

Predicted value of y at x =    11

ŷ = 26.3273 + (4.7478) * 11 = 78.55