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 x (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 13 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) SCATTER DIAGRAM OF GIVEN DATA:

(b) CALCULATION OF CORRELATION COEFFICIENT :

Let us first verify the given sums

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

The formula for sample correlation coefficient is,

  

  

Thus the sample correlation coefficient is .

(c) EQUATION OF LEAST SQUARE LINE:

I have used R code to build simple linear regression model to the given data.

Thus the equation of least square line is,

where

is the predicted dependent variable "Weight of calf"

is the intercept

is the slope coefficient of independent variable "Age of calf".

is the independent variable "Age of calf".

(d) GRAPHING THE LEAST SQUARE LINE:

(e) COEFFICIENT OF DETERMINATION :

From the above R output, the coefficent of determination is . Thus 99% of the total variation in dependent variable "weight of calf" can be explained by the corresponding variation in independent variable "Age of calf" and the least-squares line.

(f) PREDICTED WEIGHT OF CALF:

The equation of least square line is,

If the calves we want to buy are 13 weeks old, the predicted healthy weight of calf is,

  

If the calves we want to buy are 13 weeks old, the predicted healthy weight of calf is .