Question

Consider the dataset regarding the drop in light percent output (output) and two input variables – bulb surface (qualitative) and length of the operation hours (quantitative). The data is available in a sheet named ‘Problem 3’. Answer the following.

(a) Write down the overall model form if one wishes to build a second order model for each value of the qualitative variable

(c) Build a regression model showing the 90% confidence ranges of the regression parameters. Write down the mean estimates of the regression parameters for the model in (a)

(d) Write down the 90% bounds of the estimate of the y-intercept (constant term)
(e) Compute the model prediction for a bulb with a dirty surface and with 1500 operation hours.

(f) Perform one-way hypothesis testing for the existence of ?2 term, where ?2 is the qualitative input. Use ? = 0.05

DATA

Drop in light output percent (Output)	Bulb surface (Qualitative input)	Length of operation hours (Quantitative input)
0	Clean	0
16	Clean	400
22	Clean	800
27	Clean	1200
32	Clean	1600
36	Clean	2000
38	Clean	2400
0	Dirty	0
4	Dirty	400
6	Dirty	800
8	Dirty	1200
9	Dirty	1600
11	Dirty	2000
12	Dirty	2400

Answer 1

SOLUTION

(a) y=b0+b1x+b2x²

y=Drop in light output percent (Output) and x=Length of operation hours (Quantitative input)

(c)y=1.2738+0.0183*x-3.6E-06*x²

(d)

Lower 90%	Upper 90
-10.8200	13.3677

(e) for x=1500, y=1.2738+0.0183*1500-3.6E-06*1500*1500=20.6238

(f) here we use t-test and t=coefficient/SE(coefficient)=-3.6E-06/5.26E-06=-0.6859

the one-tailed critical t(0.05,11)=1.7959 is greater than absolute value of calculated t=0.6859, so x² term is not significant and it may be removed from the model ( i.e. no existence)

following regression analysis information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.637572
R Square	0.406499
Adjusted R Square	0.298589
Standard Error	10.9107
Observations	14
ANOVA
	df	SS	MS	F	Significance F
Regression	2	896.881	448.4405	3.767037	0.056730925
Residual	11	1309.476	119.0433
Total	13	2206.357
	Coefficients	Standard Error	t Stat	P-value	Lower 90%	Upper 90%
Intercept	1.27381	6.734228	0.189155	0.853417	-10.82008801	13.36770706
X	0.018348	0.013142	1.396123	0.190214	-0.005253775	0.041950204
X2	-3.6E-06	5.26E-06	-0.68591	0.506971	-1.3057E-05	5.83973E-06