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. y 0.1 0.3 2.5 160 82.9 0.2 0.2 1.5 145 83.9 0.4 0.2 1.5 145 84.9 0.5 0.3 2.5 160 85.5 0.3 0.1 2.5 160 85.2 0.2 0.4 1.5 145 83.4 0.4 | 0.4 | 15 | 145 | 84.2 0.2 0.2 3.5 145 83.8 0.40.2 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 17584.5 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 0.40.2 3.5 17586.0 0.2 0.4 3.5 17584.0 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 | 25 | 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 16084.9 0.3 0.3 2.5 160 84.5 0.3 0.3 2.5 160 84.6 0.4 0.4 3.5 175 85.4 (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.) + ]** 3+ * * + *x*, + (b) Using a 0.05 significance level, perform the model utility test. State the null and alternative hypotheses. Oll at least one of P, Pgr. or Pl is not 0. H:A-B -...-B-0 OHOR - R - - R4 - 0 M: B. By -- and B are all noto O Ho: B, B2 - ... - B1 - 0 Hat least one of B. Bor Big is not o. O Ho: B, B and , are all not HA:B - B ...-14 - 0 Calculate the test statistic. (Round your answer to two decimal places.)

Answer 1

Output using excel:

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	15.2642	1.090297	8.75	4.97E-05
Residual	16	1.9926	0.124539
Total	30	17.2568
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	76.4371	9.081796	8.416517	2.85E-07	57.18454	95.68963
x1	-7.3452	10.79946	-0.68015	0.506133	-30.2391	15.5486
x2	9.6131	10.79946	0.890146	0.386577	-13.2807	32.50693
x3	-0.9149	1.067606	-0.85695	0.404128	-3.1781	1.348342
x4	0.0963	0.098342	0.979465	0.341929	-0.11215	0.304799
x12	-13.4524	6.599354	-2.03844	0.058386	-27.4424	0.537625
x22	2.7976	6.599354	0.423923	0.677266	-11.1924	16.78763
x32	0.0280	0.065994	0.423923	0.677266	-0.11192	0.167876
x42	-0.0003	0.000293	-1.09138	0.29127	-0.00094	0.000302
x1x2	3.7500	8.82251	0.425049	0.676462	-14.9529	22.45289
x1x3	-0.7500	0.882251	-0.8501	0.407812	-2.62029	1.120289
x1x4	0.1417	0.058817	2.408612	0.028428	0.016981	0.266353
x2x3	2.0000	0.882251	2.266929	0.037608	0.129711	3.870289
x2x4	-0.1250	0.058817	-2.12525	0.049491	-0.24969	-0.00031
x3x4	0.0033	0.005882	0.566732	0.57876	-0.00914	0.015802

a)

Regression equation:

ŷ = 76.4371 + (-7.3452)x1 + (9.6131)x2 + (-0.9149)x3 + (0.0963)x4 + (-13.4524)x12 + (2.7976)x22 + (0.028)x32 + (-0.0003)x42 + (3.75)x1x2 + (-0.75)x1x3 + (0.1417)x1x4 + (2)x2x3 + (-0.125)x2x4 + (0.0033)x3x4

b)

Оно: B = Bas... = B4 = 0 Ho: at least one of B1, B2, or B14 is not 0.

F = 8.75

P-value < 0.001

Conclusion:

Reject Ho. We have convincing evidence that the multiple regression model is useful and can conclude at least one .

c)

SSResid = 1.9926

This is the sum of the squares of the deviations of the actual values from the values predicted by the fitted model.

R^2 = 0.885

This tells us the proportion of the observed variation in brightness that can be explained by the fitted model.

se = 0.3529

This is a typical deviation of a brightness value in the sample from the value predicted by the estimated regression equation.