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 19.4 16.6 20.6 17.7 16.3 15.5 14.7 17.1 y 87.4 70.8 92.7 83.1 82.6 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 = 249.1, Σy = 1198.8, Σx2 = 4176.81, Σy2 = 96,414.66, Σxy = 20,030.86, and r ≈ 0.787. (a) Draw a scatter diagram displaying the data.(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 y hat = 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 = y hat = + 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) What is the predicted temperature when x = 18.0 chirps per second? (Round your answer to two decimal places.) °F

Answer 1

Part a)

Part b)

	X	Y	X * Y	X2	Y2
	19.4	87.4	1695.56	376.36	7638.76
	16.6	70.8	1175.28	275.56	5012.64
	20.6	92.7	1909.62	424.36	8593.29
	17.7	83.1	1470.87	313.29	6905.61
	16.3	82.6	1346.38	265.69	6822.76
	15.5	75.2	1165.6	240.25	5655.04
	14.7	69.7	1024.59	216.09	4858.09
	17.1	82	1402.2	292.41	6724
	15.4	69.4	1068.76	237.16	4816.36
	16.2	83.3	1349.46	262.44	6938.89
	15	79.6	1194	225	6336.16
	17.2	82.6	1420.72	295.84	6822.76
	16	80.6	1289.6	256	6496.36
	17	83.5	1419.5	289	6972.25
	14.4	76.3	1098.72	207.36	5821.69
Total	249.1	1198.8	20030.86	4176.81	96414.66

r = 0.787

Part c)
X̅ = Σ( Xi / n ) = 249.1/15 = 16.61
Y̅ = Σ( Yi / n ) = 1198.8/15 = 79.92

Equation of regression line is Ŷ = a + bX

b = ( 15 * 20030.86 - 249.1 * 1198.8 ) / ( 15 * 4176.81 - ( 249.1 )2)
b = 3.063
a =( Σ Y - ( b * Σ X) ) / n
a =( 1198.8 - ( 3.0629 * 249.1 ) ) / 15
a = 29.056
Equation of regression line becomes Ŷ = 29.056 + 3.063 X

Part e)
Coefficient of Determination
R2 = r2 = 0.62
Explained variation = 0.62* 100 = 62%
Unexplained variation = 1 - 0.62* 100 = 38%

Part f)
When X = 18
Ŷ = 29.056 + 3.063 X
Ŷ = 29.056 + ( 3.063 * 18 )
Ŷ = 84.19