Question

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	20.5	15.5	18.8	17.5	16.3	15.5	14.7	17.1
y	87.8	70.4	92.9	83.5	81.2	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 = 247.1, Σy = 1198, Σx² = 4107.43, Σy² = 96,302.7, Σxy = 19,855.58, and r ≈ 0.796.

(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 = 19.0 chirps per second? (Round your answer to two decimal places.)
°F

Answer 1

b.

Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r.

c.

the equation of the least-squares line  = a + bx.

y = 26.015 + 3.269*x.

d.

The value of the coefficient of determination r^2 is 63.3%.

63.3 % of the variation in y can be explained by the corresponding variation in x and the least-squares line.

100 - 63.3 = 36.7% of the variation in y is unexplained by the corresponding variation in x and the least-squares line.

e.

The predicted temperature when x = 19.0 chirps per second is 88.1264.