Question

3. Description of each X and data for 27 franchise stores are given below The data (X1, X2, X3, X4, X5, X6) are for each franchise store. X1 annual net sales/$1000 X2 number sq. ft/1000 X3 - inventory I$1000 X4- amount spent on advertising /$1000 X5 size of sales district/1000 families X6 number of competing stores in distric X1 X2 X3 X4 X5 X6 231 3 294 8.2 8.2 11 156 2.2 232 6.9 4.1 12 10 0.5 149 3 4.3 15 519 5.5 600 12 16.1 1 437 4.4 567 10.6 14.1 5 487 4.8 571 11.8 12.7 4 299 3.1 512 8.1 10.1 10 195 2.5 347 7.7 8.4 12 20 1.2 212 3.3 2.1 15 68 0.6 102 4.9 47 8 570 5.4 788 17.4 12.31 428 4.2 577 10.5 14 7 464 4.7 535 11.3 15 3 15 0.6 163 2.5 2.5 14 65 1.2 168 4.7 3.3 11 98 1.6 151 4.6 2.710 398 4.3 342 5.5 16 4 161 2.6 196 7.2 6.3 13 397 3.8 453 10.4 13.9 7 497 5.3 518 11.5 16.3 1 528 5.6 615 12.3 16 0 99 0.8 278 2.8 6.5 14 0.5 1.1 142 3.1 1.6 12 347 3.6 461 9.6 11.3 6 341 3.5 382 9.8 11.5 5 507 5.1 590 12 15.7 0 400 8.6 517 7 12 8 a) Determine the most appropriate response variable and perform regression analysis with these data. Use MINITAB Use the output to answer the following questions. b) Write the fitted model. e) Is the model overall significant? Answer based on the related test. d) Is constant term significant in the model? Why? e) What is the estimated value of error variance? 0 How much of variability in response explained by the model?

Answer 1

a) The most appropriate response variable will be Annual Net Sales/$1000, rest of the variables will be predictors.

b) The fitted model estimated using MANITAB is given below

X1 = - 18.9 + 16.2 X2 + 0.175 X3 + 11.5 X4 + 13.6 X5 - 5.31 X6.

c) We will set up the null hypothesis that.

$H_{0}:\beta _{i}=0$; i= 1,2,3,4,5,6. or all regression coefficient are equal to zero.

$H_{1}:\beta _{i}\neq 0$ ; at least one regression coefficient is not equal to zero.

The overall significance of the model is used with F.Test and the Anova Table is given below

Source	DF	SS	MS	F	P.Value
Regression	5	952539	190508	611.59	0.000
Residual Error	21	6541	311
Total	26	959080

Since calculated P.value =0.000 which is less then 0.05. Hence we will accept our null hypothesis and concludes that at least one regression coefficient is not equal to zero.

d) We will set up the null hypothesis that.

Но:30 =0

$H_{1}:\beta _{0}\neq 0$

To check the individual regression coefficient is whether equal to zero or not. We use t.test.

Predictor	Coef	SE Coef	T	P.Value
Constant	-18.86	30.15	-0.63	0.538
X2	16.202	3.544	4.57	0.000
X3	0.17464	0.05761	3.03	0.006
X4	11.526	2.532	4.55	0.000
X5	13.58	1.77	7.67	0.000
X6	-5.311	1.705	-3.11	0.005

Since P.Value for Constant term is  0.538 which is greater then 0.05. hence we will accept our null hypothesis and concludes that  constant term is not significant.

It is not significant because calculated P.Value is much greater then 0.05 (significance level).

e) The estimated value of error variance = 311.

f) Since R-Sq = 99.3%. Which means that 99.3% of response variance is explained by the model.