Question

on a summer evening has probably heard crikckets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This representing chirps per second and y is a random variable re i George W. Pierce, a physics professor at Harvard. In the following data, x is a random variable representing temperature (°F) L.1 87.0 734 94.551 818 752 697 820 15.4 69.4 833 79.6 82.6 80.6835 76.3 Complete parts (a) through (e),given Ex 250.3,y 1204, 4229.21, y 97,26166, Ey 20,240.02, and r 0.822 (a) Draw a scatter diagram displaying the data Flash Player version 10 or higher is required for this question, You can get Flash Player free from Adobe's website. (b) verify the given sums rr. Ty, tr, y, rry, and the value of the sample correlation oemdent r. (Round your value for r to three decimal place.) Ex (c) Find x, and y. Then find the equation of the least-squares line +bx (Round your answers for decimal places.) and y to two decimal places, Round your answers for a and b to three i (e) Find the value of the coefficlent of 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 to three decimal places. Round your answers for the percentages to one decimal place) explained unexplained f) What is the predicted temperature when x -20.0 chirps per second? (Round your answer to two decimal places.) Need Help? iRead

Answer 1

	X	Y	X * Y	$X^2$	$Y^2$
	21	87	1827	441	7569
	15.6	73.4	1145.04	243.36	5387.56
	20.4	94.5	1927.8	416.16	8930.25
	18.3	85.1	1557.33	334.89	7242.01
	16.5	81.8	1349.7	272.25	6691.24
	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	250.3	1204	20240.02	4229.21	97261.66

Scatter Plot 100 90 80 70 60 50 40 30 20 10 14 16 18 20 10 12

$r = ( ( N * \Sigma XY) - \Sigma X \Sigma Y) / (\sqrt{ (N \Sigma (X^{2}) - (\Sigma X)^{2} )(N \Sigma (Y^{2}) - (\Sigma Y)^{2} })$
$r = ( ( 15 * 20240.02 ) - 250.3 * 1204 ) / ( \sqrt{ ( 15 * 4229.21 ) - ( 250.3 )^{2} )( 15 * 97261.66 ) - ( 1204 )^{2} })$
r = 0.827

$\bar{X} = \Sigma X_{i} / n = 250.3/15 = 16.69$
$\bar{Y} = \Sigma Y_{i} / n = 1204/15 = 80.27$

Equation of regression line is $\hat{Y} = a + bX$
$b = ( n \Sigma XY - \Sigma X \Sigma Y) / ( n\Sigma X^{2} - (\Sigma X)^{2})$

b = 2.841

$a =( \Sigma Y - ( b * \Sigma X) ) / n$
a =( 1204 - ( 2.8413 * 250.3 ) ) / 15
a = 32.855
Equation of regression line becomes $\hat{Y} = 32.8552 + 2.8413X$

Coefficient of Determination

Explained variation = 0.683* 100 = 68.3%
Unexplained variation = 1 - 0.683* 100 = 31.7%

When X = 20
$\hat{Y}$ = 32.855 + 2.841 X
$\hat{Y}$ = 32.855 + 2.841 * 20
$\hat{Y}$ = 89.68