Question

The accompanying data resulted from a study of the relationship between y = brightness of finished paper and the independent variables

x₁ = hydrogen peroxide (% by weight), x₂ = sodium hydroxide (% by weight), x₃ = silicate (% by weight), and x₄ = process temperature.

x1	x2	x3	x4	y
0.2	0.2	1.5	145	83.9
0.4	0.2	1.5	145	84.9
0.2	0.4	1.5	145	83.4
0.4	0.4	1.5	145	84.2
0.2	0.2	3.5	145	83.8
0.4	0.2	3.5	145	84.7
0.2	0.4	3.5	145	84.0
0.4	0.4	3.5	145	84.8
0.2	0.2	1.5	175	84.5
0.4	0.2	1.5	175	86.0
0.2	0.4	1.5	175	82.6
0.4	0.4	1.5	175	85.1
0.2	0.2	3.5	175	84.5
0.4	0.2	3.5	175	86.0
0.2	0.4	3.5	175	84.0
0.4	0.4	3.5	175	85.4

x1	x2	x3	x4	y
0.1	0.3	2.5	160	82.9
0.5	0.3	2.5	160	85.5
0.3	0.1	2.5	160	85.2
0.3	0.5	2.5	160	84.5
0.3	0.3	0.5	160	84.7
0.3	0.3	4.5	160	85.0
0.3	0.3	2.5	130	84.9
0.3	0.3	2.5	190	84.0
0.3	0.3	2.5	160	84.5
0.3	0.3	2.5	160	84.7
0.3	0.3	2.5	160	84.6
0.3	0.3	2.5	160	84.9
0.3	0.3	2.5	160	84.9
0.3	0.3	2.5	160	84.5
0.3	0.3	2.5	160	84.6

(a) Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. (Round your answers to four decimal places.)

= ____ + _____ x₁ + ____ x₂ + ____ x₃ + ____ x₄ + _____ x₁² + _____ x₂² + ______ x₃² + _____ x₄² + ______ x₁x₂ + _____ x₁x₃ + _____ x₁x₄ + _____ x₂x₃ + ______ x₂x₄ + ______ x₃x₄

(b) Calculate SSResid. (Round your answer to four decimal places.)

SSResid =  

Answer 1

using excel>data>data analysis>Regression

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.940495
R Square	0.884531
Adjusted R Square	0.783496
Standard Error	0.3529
Observations	31
ANOVA
	df	SS	MS	F	Significance F
Regression	14	15.26416	1.090297	8.754683	4.97E-05
Residual	16	1.992619	0.124539
Total	30	17.25677
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	76.43709	9.081796	8.416517	2.85E-07	57.18454	95.68963	57.18454	95.68963
x1	-7.34524	10.79946	-0.68015	0.506133	-30.2391	15.5486	-30.2391	15.5486
x2	9.613095	10.79946	0.890146	0.386577	-13.2807	32.50693	-13.2807	32.50693
x3	-0.91488	1.067606	-0.85695	0.404128	-3.1781	1.348342	-3.1781	1.348342
x4	0.096323	0.098342	0.979465	0.341929	-0.11215	0.304799	-0.11215	0.304799
X12	-13.4524	6.599354	-2.03844	0.058386	-27.4424	0.537625	-27.4424	0.537625
X22	2.797619	6.599354	0.423923	0.677266	-11.1924	16.78763	-11.1924	16.78763
X32	0.027976	0.065994	0.423923	0.677266	-0.11192	0.167876	-0.11192	0.167876
X42	-0.00032	0.000293	-1.09138	0.29127	-0.00094	0.000302	-0.00094	0.000302
X1X2	3.75	8.82251	0.425049	0.676462	-14.9529	22.45289	-14.9529	22.45289
X1X3	-0.75	0.882251	-0.8501	0.407812	-2.62029	1.120289	-2.62029	1.120289
X1X4	0.141667	0.058817	2.408612	0.028428	0.016981	0.266353	0.016981	0.266353
X2X3	2	0.882251	2.266929	0.037608	0.129711	3.870289	0.129711	3.870289
X2X4	-0.125	0.058817	-2.12525	0.049491	-0.24969	-0.00031	-0.24969	-0.00031
X3X4	0.003333	0.005882	0.566732	0.57876	-0.00914	0.015802	-0.00914	0.015802

a) the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms is

= 76.4371 -7.3452 x1 + 9.6131 x2 -0.9149 x3 +0.0963 x4 -13.4524 x12 +2.7976 x22 +0.0280 x32 -0.0003 x42 + 3.7500 x1x2 -0.7500 x1x3 + 0.1416 x1x4 + 2.000 x2x3 -0.1250 x2x4 +0.0033 x3x4

(b)

SSResid =  1.9926