Question

The data in stat4_prob5 present the performance of a chemical process as a function of sever controllable process variables.

1. (a) Fit a multiple regression modelrelating CO2product (y) to total solvent (x1) and hydrogen consumption (x2) and report the fitted regression line.
2. (b) Find a point estimatefor the variance term σ2.
3. (c) Construct the ANOVA tableand test for the significance of the regression using

α = 0.05.

4. (d) Using individual t-tests determine the contribution of x1 and x2 to the model using α = 0.05.

Here is the data:

y=c(36.98, 13.74, 10.08, 8.53, 36.42, 26.59, 19.07, 5.96, 15.52, 56.61, 26.72, 20.80, 6.99, 45.93, 43.09, 15.79, 21.60, 35.19, 26.14, 8.60, 11.63, 9.59, 4.42, 38.89, 11.19, 75.62, 36.03)

x1=c(2227.25, 434.90, 481.19, 247.14, 1645.89, 907.59, 608.05, 380.55, 213.40, 2043.36, 761.48, 566.40, 237.08, 1961.49, 1023.89, 411.30, 2244.77, 978.64, 687.62, 468.28, 460.62, 290.42, 233.95, 2088.12, 994.63, 2196.17, 1080.11)

x2=c(2.06, 1.33, 0.97, 0.62, 0.22, 0.76, 1.71, 3.93, 1.97, 5.08, 0.60, 0.90, 0.63, 2.04, 1.57, 2.38, 0.32, 0.44, 8.82, 0.02, 1.72, 1.88, 1.43, 1.35, 1.61, 4.78, 5.88)

Answer 1

excel regression output is

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.836448011
R Square	0.699645274
Adjusted R Square	0.674615714
Standard Error	9.92435002
Observations	27
ANOVA
	df	SS	MS	F	Significance F
Regression	2	5506.286737	2753.143368	27.9527591	5.39031E-07
Residual	24	2363.82536	98.49272332
Total	26	7870.112096
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	2.526630302	3.610029262	0.699891917	0.4907239	-4.924103899	9.977364504	-4.924103899	9.977364504
X1	0.018521965	0.002747236	6.74203717	5.66264E-07	0.012851949	0.024191981	0.012851949	0.024191981
X2	2.185719512	0.972694428	2.247077243	0.034100063	0.178176882	4.193262142	0.178176882	4.193262142

a)

fitted regression line is

Y hat =    2.5266 +   0.0185 *X1+   2.1857 *X2

b)

point estimatefor the variance term σ2.=Se^2 = 9.92435002^2 = 98.493

c)

ANOVA
	df	SS	MS	F	Significance F
Regression	2	5506.286737	2753.143368	27.9527591	5.39031E-07
Residual	24	2363.82536	98.49272332
Total	26	7870.112096

d)

for X1

Ho:ß1=0

Ha:ß1╪0

t-value=6.74203717

pvalue=0.000<alpha=0.05,reject Ho

so, slope test is significant

---------

for X2

Ho:ß2=0

Ha:ß2╪0

t-value=2.2471

p-value=0.0341<alpha=0.05,reject Ho

so, slope test is significant