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.)

A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (1, 29), (5, 37), (11, 63), (16, 90), (26, 140), (36, 190). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right.

A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (1, 39), (5, 47), (11, 73), (16, 100), (26, 150), (36, 200). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right.

A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (3, 29), (7, 37), (13, 63), (18, 90), (28, 140), (38, 190). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right.

A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (3, 39), (7, 47), (13, 73), (18, 100), (28, 150), (38, 200). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right.

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

x bar = x bar =

y bar = y bar =

y hat = value of a coefficient + value of b coefficient x

(d)

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

A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 102). The line enters the window at approximately y = 26 on the positive y axis, goes up and right, passes through the approximate point (15.8, 102), and exits the window in the first quadrant.

A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 102). The line enters the window at approximately y = 171 on the positive y axis, goes down and right, passes through the approximate point (15.8, 102), and exits the window at approximately x = 39.1 on the positive x axis.

A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 132). The line enters the window at approximately y = 201 on the positive y axis, goes down and right, passes through the approximate point (15.8, 132), and exits the window in the first quadrant.

A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 132). The line enters the window at approximately y = 56 on the positive y axis, goes up and right, passes through the approximate point (15.8, 132), and exits the window in the first quadrant.

(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 14 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

I used R software to solve this problem.

R codes:

x=c(1,5,11,16,26,36)
y=c(39,47,73,100,150,200)
plot(x,y)

a. Scatter plot

Option 2 is correct.

A scatter diagram with 6 points is graphed on the x y coordinate plane. The points are located at (1, 39), (5, 47), (11, 73), (16, 100), (26, 150), (36, 200). When considering the data points as a whole, the points are loosely gathered into a mass that is lower on the left and higher on the right.

b.

> sum(x)
[1] 95
> sum(y)
[1] 609
> sum(x^2)
[1] 2375
> sum(y^2)
[1] 81559
> sum(x*y)
[1] 13777
> cor(x,y)
[1] 0.9970603

All values are same as given values.

Σx = 95

Σy = 609

Σx2 = 2375

Σy2 = 81559

Σx y = 13777

r = 0.997

c.

> mean(x)
[1] 15.83333
> mean(y)
[1] 101.5
> fit=lm(y~x)
> fit

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept) x
26.327 4.748

x bar = 15.83

y bar = 101.50

y hat = value of a coefficient + value of b coefficient x = 26.327 + 4.748 x

d.

A line and a point are graphed on the x y coordinate plane. The point is located the approximate point (15.8, 102). The line enters the window at approximately y = 26 on the positive y axis, goes up and right, passes through the approximate point (15.8, 102), and exits the window in the first quadrant

e.

> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5 6
7.9250 -3.0660 -5.5525 -2.2913 0.2312 2.7537

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 26.3273 3.6295 7.254 0.00192 **
x 4.7478 0.1824 26.026 1.3e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.383 on 4 degrees of freedom
Multiple R-squared: 0.9941, Adjusted R-squared: 0.9927
F-statistic: 677.3 on 1 and 4 DF, p-value: 1.295e-05

r2= 0.994

Explained variation = 99.4%

Unexplained variation = 100 - 99.4 =0.6 %

f.

If x = 14 then Y can be obtained as follows,

y= 26.327 + 4.748 x

= 26.327 + 4.748 * 14

= 92.799