Question

11.7. Machine speed. The number of defective items produced by a machine (Y) is known to be linearly related to the speed setting of the machine (X). The data below were collected from recent quality control records. #d. Estimate the variance function by regressing the squared residuals against-X, and then calculate the estimated weight for each case using (11.16b). #e. Using the estimated weights, obtain the weighted least squares estimates of f30 and th. Are the weighted least squares estimates similar to the ones obtained with ordinary least squares in part (a)? #f. Compare the estimated standard deviations of the weighted least squares estimates bwo and bW1 in part (e) with those for the ordinary least squares estimates in part (a). What do you find? #g. Iterate the steps in parts (d) and (e) one more time. Is there a substantial change in the estimated regression coefficients? If so, what should you do?

   28.0 200.0
   75.0 400.0
   37.0 300.0
   53.0 400.0
   22.0 200.0
   58.0 300.0
   40.0 300.0
   96.0 400.0
   46.0 200.0
   52.0 400.0
   30.0 200.0
   69.0 300.0

Answer 1

Line of Regression Y on X i.e Y = bo + b1 X
					X	Y	(Xi - Mean)^2	(Yi - Mean)^2	(Xi-Mean)*(Yi-Mean)
28	200	506.25	10000	2250
75	400	600.25	10000	2450
37	300	182.25	0	0
53	400	6.25	10000	250
22	200	812.25	10000	2850
58	300	56.25	0	0
40	300	110.25	0	0
96	400	2070.25	10000	4550
46	200	20.25	10000	450
52	400	2.25	10000	150
30	200	420.25	10000	2050
69	300	342.25	0	0

calculation procedure for regression

mean of X = sum ( X / n ) = 50.5

mean of Y = sum ( Y / n ) = 300

sum ( (Xi - Mean)^2 ) = 5129

sum ( (Yi - Mean)^2 ) = 80000

sum ( (Xi-Mean)*(Yi-Mean) ) = 15000

b1 = sum ( (Xi-Mean)*(Yi-Mean) ) / sum ( (Xi - Mean)^2 )

= 15000 / 5129

= 2.925

bo = sum ( Y / n ) - b1 * sum ( X / n )

bo = 300 - 2.925*50.5 = 152.31

value of regression equation is, Y = bo + b1 X

Y'=152.31+2.925* X

bo =152.31

b1 =2.925

Standard Error of Y on X i.e Y = bo + b1 X
					Xi	Yi	Y'=152.31+2.92*X	Y-Y'	(Y-Yi)^2
28	200	234.21	-34.21	1170.324
75	400	371.685	28.315	801.739
37	300	260.535	39.465	1557.486
53	400	307.335	92.665	8586.802
22	200	216.66	-16.66	277.556
58	300	321.96	-21.96	482.242
40	300	269.31	30.69	941.876
96	400	433.11	-33.11	1096.272
46	200	286.86	-86.86	7544.66
52	400	304.41	95.59	9137.448
30	200	240.06	-40.06	1604.804
69	300	354.135	-54.135	2930.598

Standard error = Sqrt( ( sum ( Y -Yi )^2/ n-2 )

sum ( Y -Yi )^2 = 36131.807

Standard Error = 36131.807