Question

Problem 3.

A department store investigated the effects of advertising expenditure on the weekly sales for its men's wear, children's wear, and women's wear departments. Five weeks were randomly selected for each department to be used in the analysis (this makes 15 weeks in total, ?n). The variables are as follows:

?y = weekly sales
?1x1 = advertising expenditure
?2x2 = 1 if it is the children's wear department and a 0 otherwise
?3x3 = 1 if it is the women's wear department and a 0 otherwise

The manager in charge of the department store wants to see if separating by the type of department is significant to the model. The current model is:
?̂=?0+?1?1+?2?2+?3?3y^=β0+β1x1+β2x2+β3x3

a) Test whether the part of the model which concerns the departments is significant or not using ?=0.05α=0.05 and the tables below.

Complete model:

Source	df	SS	MS	F
Regression	3	70.193	23.398	16.341
Error	11	15.751	1.432
Total	14	85.944

Reduced model:

Source	df	SS	MS	F
Regression	1	0.161	0.161	0.024
Error	13	85.783	6.599
Total	14	85.944

1. ?0:H0:   B0 = B1 = B2 = B3 = 0 B1 = B2 = B3 = 0 B2 = B3 = 0 B1 = B2 = 0 B0 = 0 B1 = 0 B2 = 0 B3 = 0  vs ??:Ha:   B1 /= 0 At least one of the B's are not 0 B2 /= 0 B3 /= 0 B0 /= 0  
2. Test statistic: F =  
3. Critical value: ??Fα =  
4. Conclusion:  ?0H0. (Type either Reject or Fail to reject)
There  evidence that the part of the model which concerns the departments is significant. (Type either is or is not)

b) What should the model be now?
?̂=?0+?1?1+?2?2+?3?3y^=β0+β1x1+β2x2+β3x3
?̂=?0+?1?1y^=β0+β1x1
?̂=?0+?1?1+?2?2y^=β0+β1x1+β2x2
?̂=?0+?1?1+?2?3

Answer 1

a)F stat=16.341

Critical Fstat=F(0.05,3,11)=8.76

As Fstat>8.76, we shall reject H0 and Hence, atleast one of the coeff is non zero.

b)Fstat=0.024

Critical Stat=F(0.05,1,13)=0.00048

Hence as Fstat>critical, we reject H0 and say ateast one of beta is non zero.

Things to be noted are that the dof on the error line is 1 which means that the eqn with one independent variable would be ans

Ans->?̂=?0+?1?1y^=β0+β1x1