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.

(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 xand 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 15 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

a) -

b) -

Observation table -

Sr. No.	X	Y	X^2	Y^2	XY
1	1	39	1	1521	39
2	5	47	25	2209	235
3	11	73	121	5329	803
4	16	100	256	10000	1600
5	26	150	676	22500	3900
6	36	200	1296	40000	7200
Total	95	609	2375	81559	13777

Formula for correlation coefficient -

  

Hence, correlation coefficient r is 0.997.

c) -

Least square regression line is as follows -

Y = a + bX

Where,

a = - b

Calculations -

  

a = - b = 101.5 - (4.748)(15.83) = 101.5 - 75.1608 = 26.3392 26.339

Value of coefficients a & b are -

a = 26.39

b = 4.748

Regression equation becomes -

Y = 26.339 + 4.748X

d) -

e) -

Coefficient of determination, r² = (0.997)² = 0.994

99.4% of variation explained by the corresponding variation in x & least square line. 0.6% variation is unexplained.

Explained = 99.4%

Unexplained = 0.6%

f) -

The calves want to buy are 15 weeks old. i.e. X = 15

Hence, by using regression equation -

Y = 26.339 + 4.748X = 26.339 + 4.748(15) = 26.339 + 71.22 = 97.559 97.56

The least square line predicts for healthy weight about 97.56 kg.