Question

a) Obtain the equation of the least squares line.

b) Calculate the coefficient of determination.

c) Calculate an estimate of the error standard deviation (sigma) in the simple linear regression model.

Your T For the past decade, rubber powder has been used In asphalt cement to improve performance. An article Includes a (MPa) on x - cube strength (MPa) based on the following sample data regression of y-axial strength x112.3 97.0 92.7 86.0 102.0 99.2 95.8 103.5 89.0 86.7 y| 74.5 71.4 57.6 49.2 74.5 73.4 68.1 59.2 57.7 48.0 (a) Obtain the equation of the least squares line. (Round all numerical values to four decimal places.) Interpret the slope O A one MPa decrease in axial strength is assoclated with an increase in the predicted cube strength equal to the slope. In ax O A one MPa increase in cube strength is associated with an increase in the predicted axial strength equal to the slope. O A one MPa decrease in cube strength is associated with an increase in the predicted axial strength equal to the slope. (b) Calculate the coefficient of determination. (Round your answer to four decimal places)

Answer 1

Following table shows the calculations :

	X	Y	X^2	Y^2	XY
	112.3	74.5	12611.29	5550.25	8366.35
	97	71.4	9409	5097.96	6925.8
	92.7	57.6	8593.29	3317.76	5339.52
	86	49.2	7396	2420.64	4231.2
	102	74.5	10404	5550.25	7599
	99.2	73.4	9840.64	5387.56	7281.28
	95.8	68.1	9177.64	4637.61	6523.98
	103.5	59.2	10712.25	3504.64	6127.2
	89	57.7	7921	3329.29	5135.3
	86.7	48	7516.89	2304	4161.6
Total	964.2	633.6	93582	41099.96	61691.23

Sample size: n =  10

Now,

$S_{yy}=\sum y^{2}-\frac{\left (\sum y \right )^{2}}{n}=955.064$

$S_{xx}=\sum x^{2}-\frac{\left (\sum x \right )^{2}}{n}=613.836$

$S_{xy}=\sum xy-\frac{\left (\sum x \right )\left (\sum y \right )}{n}=599.518$

(A)

Slope of the regression equation is

$b_{1}=\frac{S_{xy}}{S_{xx}}=0.9767$

and intercept of the equation will be

$b_{0}=\frac{1}{n}(\sum y - b_{1} \sum x)=-30.8110$

So the regression equation will be

y'= -30.8110+0.9767x

(b)

$SSR=\frac{S^{2}_{xy}}{S_{xx}}=585.5339738$
$SSE=S_{yy}-\frac{S^{2}_{xy}}{S_{xx}}=369.5300262$

$SST=955.064$

The coefficient of determination is

$R^{2}=\frac{SSR}{SST}=0.6131$

The estimate of the error standard deviation (sigma) in the simple linear regression model is

$\sigma=\sqrt{\frac{SSE}{n-2}}=\sqrt{\frac{369.5300262}{8}}=6.7964$