Question

4. Comparing the fit of the regression lines for two sets of data Aa Aa E Examine each of the following scatter diagrams and the corresponding regression lines. Identify which line better fits its data. Graph I Graph 11 Next, calculate a measure of how close the data points are to the regression line. Following are the six pairs of data values for Graph I, along with the regression equation: 5.6 6.6 9.6 y = -0.25 + 1.44x

Answer 1

The 2nd image is a better fit, as the observed points are very close to regression line as opposed to the 1st image.

For Graph I,
The regression line is given as:
$\widehat{y}=-0.25+1.44x$

We predict on the given data, using the given regression estimate.
Using the above equation, we get the following table:

xi	yi	$\widehat{y_i}$	$y_i-\widehat{y_i}$	$(y_i-\widehat{y_i})^2$
1	2.6	1.19	1.41	1.9881
2.2	0.2	2.92	-2.72	7.3984
3.4	7	4.65	2.35	5.5225
4.4	4.4	6.09	-1.69	2.8561
5.6	8.2	7.81	0.39	0.1521
6.6	9.6	9.25	0.35	0.1225
			Total	18.0397

Thus, the SSE is 18.0397

The corrosponding degrees of freedom is 6-2 = 4.

The Standard Error of the Estimate =$\sqrt{\frac{SSE}{N}}=\sqrt{\frac{18.0397}{6}}=1.73396$

For Graph II,
The regression line is given as:
$\widehat{y}=1.44x+0.23$

We predict on the given data, using the given regression estimate.
Using the above equation, we get the following table:

xi	yi	$\widehat{y_i}$	$y_i-\widehat{y_i}$	$(y_i-\widehat{y_i})^2$
1	1.6	1.67	-0.07	0.0049
2.2	3.4	3.4	0	0
3.4	5.2	5.13	0.07	0.0049
4.4	6.4	6.57	-0.17	0.0289
5.6	8.8	8.29	0.51	0.2601
6.6	9.4	9.73	-0.33	0.1089
			Total	0.4077

Thus, the SSE is 0.4077

The corrosponding degrees of freedom is 6-2 = 4.


The Standard Error of the Estimate =$\sqrt{\frac{SSE}{N}}=\sqrt{\frac{0.4077}{6}}=0.06795$



Thus, the Standard Error of estimate for Graph I is more than that of Graph II.
Thus, the least square lie for Graph I fits the data less efficiently than the least square line for Graph II fits its' data.
The magnitude of the correlation cofficient for Graph i to be less than the correlation coefficient for Graph II.



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