Question

: Carbon dioxide CO2 is produced by burning fossil fuels such as oil and natural gas, and has been connected to global warming. The following table presents the average amounts (in metric tons) of

CO2 emissions for certain years per person in the United States and per person in the rest of the world.

Year	Non-U.S.	U.S.
1992	3.5	18.8
1993	3.3	19.7
1995	3.3	19.5
1996	3.4	19.7
2001	3.3	19.6
2002	3.3	19.6
2003	3.5	19.4
2004	3.7	19.6
2005	3.8	19.5
2006	3.7	19.0

Compute the least-squares regression line for predicting U.S. emissions from non-U.S. emissions. Round the slope and

y-intercept to at least four decimal places.

Answer 1

The following data are :

Non-U.S	U.S.
3.5	18.8
3.3	19.7
3.3	19.5
3.4	19.7
3.3	19.6
3.3	19.6
3.5	19.4
3.7	19.6
3.8	19.5
3.7	19.0

The independent variable is Non−U.S, and the dependent variable is U.S.. In order to compute the regression coefficients, the following table needs to be used:

	Non-U.S	U.S.	Non-U.S*U.S.	Non-U.S2	U.S.2
	3.5	18.8	65.8	12.25	353.44
	3.3	19.7	65.01	10.89	388.09
	3.3	19.5	64.35	10.89	380.25
	3.4	19.7	66.98	11.56	388.09
	3.3	19.6	64.68	10.89	384.16
	3.3	19.6	64.68	10.89	384.16
	3.5	19.4	67.9	12.25	376.36
	3.7	19.6	72.52	13.69	384.16
	3.8	19.5	74.1	14.44	380.25
	3.7	19.0	70.3	13.69	361
Sum =	34.8	194.4	676.32	121.44	3779.96

Based on the above table, the following is calculated:

$\bar X = \frac{1}{n} \sum_{i=1}^{n} X_i = \frac{ 34.8}{ 10} = 3.48$

$\bar Y = \frac{1}{n} \sum_{i=1}^{n} Y_i = \frac{ 194.4}{ 10} = 19.44$

$SS_{XX} = \sum_{i=1}^{n} X_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right)^2 = 121.44 - 34.8^2/10 = 0.33599999999997$$SS_{YY} = \sum_{i=1}^{n} Y_i^2 - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} Y_i\right)^2 = 3779.96 - 194.4^2/10 = 0.82399999999961$$SS_{XY} = \sum_{i=1}^{n} X_i Y_i - \displaystyle\frac{1}{n}\left(\sum_{i=1}^{n} X_i\right) \left(\sum_{i=1}^{n} Y_i\right) = 676.32 - 34.8 \times 194.4/10 = -0.19200000000012$

The regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

$m = \frac{SS_{XY}}{SS_{XX}} = \frac{ -0.19200000000012}{ 0.33599999999997} = -0.5714$

$n = \bar Y - \bar X \cdot m = 19.44 - 3.48 \times (-0.5714) = 21.4286$

Therefore, we find that the regression equation is:

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