Question

Answer 1

a) Scatter plot of Power consumption vs Avg Product Purity is shown below. The relationship looks non-linear from the plot below.

95 94 93 92 90 89 o 88 86 85 50 100 150 250 300 350 X1 (AvgProdPurity)

b) Correlation(Y, X1) = 0.0488

Correlation(Y, X2) = -0.0092

Looking at the low correlation coefficient in either case, both the independent variables show a weak linear relationship with the dependent variable.

c) Carrying out regression between Y, X1 and X2 in excel (go to Data tab-> Data Analysis -> Regression) we get the following output:

Regression Statistics
Multiple R	0.050571764
R Square	0.002557503
Adjusted R Square	-0.219096385
Standard Error	27.16048318
Observations	12

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	235.4441624	337.9871826	0.696606778	0.503641767	-529.1359637	1000.024288
X1 (AvgProdPurity)	0.540644691	3.619346314	0.149376336	0.884550424	-7.646885498	8.72817488
X2 (TonsProduced)	-0.050252565	1.27775449	-0.039328811	0.969486837	-2.940734037	2.840228907

Hence, least squares regression line: Y = 235.44 + 0.541 * X1 - 0.0503 * X2

d) Coefficient of purity = 0.541,which is +ve suggesting a positive linear relationship with power consumed. However, the high p-value of 0.8845 suggests this correlation is weak

Coefficient of purity = -0.0503,which is +ve suggesting a negative linear relationship with power consumed. However, the high p-value of 0.9695 suggests this correlation is again very weak

e) Coefficient of determination, R-squared = 0.00256

=> only 0.256% of the variation in Y (Power consumption) is explained by the variation in X1 and X2

f) Given p-values for both avg product purity (X1) and tons of product produced (X2) are high (>>0.05), they are not useful predictors of power consumption

g) The p-value of both the coefficients is extremely high (0.8845 and 0.9695), much higher than 0.01, suggesting it is a very poor linear regression model where both the predictor variables fail in predicting the dependent variable at any significant level of accuracy.

h) For X1 = 90% and X2 = 98, we have

Y = 235.44 + 0.541 * 90 - 0.0503 * 98 = 279.2 (1000 kWh)

i) The assumption of the fitted model is that the errors in prediction are normally distributed, which doesn't look like the case from the prediction errors (Y_predicted - Y_observed) plotted below, which is far from normal, thus violating the most basic premise of regression.

Error(Pred) 50 40 漠) 20 10 9 10 12 4 10 20 漠) 4