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 call weigh? Let x be the age of the call (in weeks), and let y be the weight of the calf (in Ilograms). X y 3 40 16 100 74 20 150 30 200 54 Verify the given sure Ix, Ly, Ix2, 3y2 Exy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) 5.72 Find, and then find the equation of the least-squares line = a + box. (Round your answers for and to two decimal places. Round your answers for a and b to three decimal places.) X- y = Graph the least squares line. Be sure to plot the point) as a point on the line y 2001 2001 150 150 100 100 50 50 0 X 10 20 SO 40 10 20 30 40 у у 200H 200 150 1500 100 100 50F 50 0 O 10 20 30 40 20 301 40 Find the value of the coefficient or determination 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 2 to three decimal places Round your answers for the percentages to one decimal place.) explained unexplained 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

х Y 3 40 2 54 9 74 16 100 26 150 36 200 The independent variable is X, and the dependent variable is Y. In order to compute the regression coefficients, the following table needs to be used: х Y X*Y x2 y2 3 40 120 9 1600 2 54 108 4 2916 9 74 666 81 5476 16 100 1600 256 10000 26 150 3900 676 22500 36 200 7200 1296 40000 Sum = 92 618 13594 2322 82492

So ,
2
= 92 , $\sum y$ = 618 ,  $\sum x^2$= 2322, $\sum y^2$ =82492 , $\sum xy$ = 13594

For finding correlation coefficient r we further calculate followings:

The correlation coefficient r is computed using the following expression: r = SSXY SSXX SSYY where SSxy = x;Y; - i=1 ΣΧΥ, : (Ex) (Y) -Ex?-? (3x) SSxx 2 n 1 SSYY ΣΥ? Y; n i=1 i=1 In this case, based on the data provided, we get that SSxy = 13594 1 (92 x 618) = 6 4118 SSxx = 2322 – (92) = 911.333 (618) 1 SSyy = 82492 18838 6 Therefore, based on this information, the sample correlation coefficient is computed as follows r= SSXY SSxx SSyy 4118 911.333 x 18838 = 0.994

Thus r = 0.994

Value of
r
and $\bar{y}$ are as follows:

For the Regression Line :

Based on the above table, the following is calculated: n 1 92 X X; = 15.333333333333 n 6 i=1 72 1 Y - = ΣΥ, = 618 6 = 103 п i=1

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y- intercept n) are obtained as follows: m = SSXY SSxx = 4118 911.33333333333 4.5187 n=Y – X.m = 103 – 15.333333333333 x 4.5187 = 33.714 Therefore, we find that the regression equation is: Y = 33.714 + 4.5187X

Therefore, $\widehat{y} = 33.714+4.5189*x$

Plot of the Line:

Scatter Plot and Regression Line 208 158 Y 108 58 --4.80 8.20 5.20 10.20 15.20 20.20 25.20 30.20 35.20 40.20 Regression equation: Y = 33.714 + 4.5187*X

As intercept is 33.714 Therefore the 3rd plot is the right answer for the line.

Coefficient of Determination = (correlation coefficient)² = 0.994² = 0.988

Explained variation = (r²)*100 % = 98.8%
Unexplained variation = (1-r²) *100% = 1.2%

Healthy weight (y) for age 24 weeks(x) = 33.714 + 4.5187*24 = 142.16