Question

9.2

1)

Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following data, x is a random variable representing chirps per second and y is a random variable representing temperature (°F).

x	19.1	16.1	19.2	18.1	17.2	15.5	14.7	17.1
y	90.2	72.4	93.3	85.1	82.0	75.2	69.7	82.0

x	15.4	16.2	15.0	17.2	16.0	17.0	14.4
y	69.4	83.3	79.6	82.6	80.6	83.5	76.3

Complete parts (a) through (e), given Σx = 248.2, Σy = 1205.2, Σx² = 4137.66, Σy² = 97,490.3, Σxy = 20,063.68, and r ≈ 0.856.

(b) Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

Σx =
Σy =
Σx2 =
Σy2 =
Σxy =
r =

(c) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x	=
y	=
	=	+  x

(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? (Round your answer for r² to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained	%
unexplained	%

(f) What is the predicted temperature when x = 19.4 chirps per second? (Round your answer to two decimal places.)
°F

2)

(a) Suppose you are given the following (x, y) data pairs.

x	2	3	5
y	4	3	9

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =  +   x

(b) Now suppose you are given these (x, y) data pairs.

x	4	3	9
y	2	3	5

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =   +   x

(c) In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair?

YesNo   

(d) Solve your answer from part (a) for x (rounded to three digits after the decimal).
x =   +   y

3)

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	3	4	12	16	26	36
y	42	54	70	100	150	200

Complete parts (a) through (e), given Σx = 97, Σy = 616, Σx² = 2397, Σy² = 82,080, Σxy = 13,882, and r ≈ 0.993.

(b) Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

Σx =
Σy =
Σx2 =
Σy2 =
Σxy =
r =

(c) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x	=
y	=
	=	+  x

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

(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? (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 22 weeks old. What does the least-squares line predict for a healthy weight? (Round your answer to two decimal places.)
kg

Answer 1

1) b.

X Values
∑ = 137
Mean = 17.125
∑(X - Mx)2 = SSx = 18.735

Y Values
∑ = 649.9
Mean = 81.238
∑(Y - My)2 = SSy = 489.579

X and Y Combined
N = 8
∑(X - Mx)(Y - My) = 93.382

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = 93.382 / √((18.735)(489.579)) = 0.975

c.

Sum of X = 137
Sum of Y = 649.9
Mean X = 17.125
Mean Y = 81.2375
Sum of squares (SSX) = 18.735
Sum of products (SP) = 93.3825

Regression Equation = ŷ = bX + a

b = SP/SSX = 93.38/18.74 = 4.984

a = MY - bMX = 81.24 - (4.98*17.13) = -4.120

ŷ = 4.984X - 4.120

e. Here r=0.975

So r^2=0.975^2=0.951

So explained variation is 95.1%

Unexplained variation is 4.9%

f. for x=19.4, ŷ = (4.984*19.4) - 4.120=92.57