Question

6. Concerns about global warming have led to studies of the relationship between global temperature and the concentration of carbon dioxide (in parts per million) and temperature in Celsius for different years. Using the chart below is there a linear correlation between temperature and concentrations of carbon dioxide? Explain your reasoning (6 points) CO2 Temp. 314 13.9 317 14 3 20 13.9 326 14.1 331 14 339 14,3 346 14.1 354 14.5 361 14.5 7. Refer to question #6. If you were able to prove a significant correlation, give the least squares regression equation. (6 points) 8. Refer to Question #7 and find the residual for a CO2 level of 320. (8 points)

Answer 1

	X	Y	X * Y	X2	Y2	Ŷ	Residual = ( Y - Ŷ )
	314	13.9	4364.6	98596	193.2100	13.8888	0.0112
	317	14	4438	100489	196.0000	13.9267	0.0733
	320	13.9	4448	102400	193.2100	13.9647	-0.0647
	326	14.1	4596.6	106276	198.8100	14.0405	0.0595
	331	14	4634	109561	196	14.1037	-0.1037
	339	14.3	4847.7	114921	204.4900	14.2048	0.0952
	346	14.1	4878.6	119716	198.8100	14.2933	-0.1933
	354	14.5	5133	125316	210.2500	14.3944	0.1056
	361	14.5	5234.5	130321	210.2500	14.4829	0.0171
Total	3008	127.3	42575	1007596	1801.0300	24708.1314

Part 7

$r = ( ( N * \Sigma XY) - \Sigma X \Sigma Y) / (\sqrt{ (N \Sigma (X^{2}) - (\Sigma X)^{2} )(N \Sigma (Y^{2}) - (\Sigma Y)^{2} })$
$r = ( ( 9 * 42575 ) - 3008 * 127.3 ) / ( \sqrt{ ( 9 * 1007596 ) - ( 3008 )^{2} )( 9 * 1801.03 ) - ( 127.3 )^{2} })$
r = 0.9027

There is strong and positive correlation between variables

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X2 - (ΣX)2 )
b = ( 9 * 42575 - 3008 * 127.3 ) / ( 9 * 1007596 - ( 3008 )2)
b = 0.0126

a =( ΣY - ( b * ΣX ) ) / n
a =( 127.3 - ( 0.0126 * 3008 ) ) / 9
a = 9.9197
Equation of regression line becomes Ŷ = 9.9197 + 0.0126 X

Part 8

Residual

For CO2 = 320

Predicted value Ŷ =   13.9647

Residual   ( Y - Ŷ ) =   -0.0647