Question

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?3

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	41.268	13.756	9.067
Error	11	16.689	1.517
Total	14	57.957

Reduced model:

Source	df	SS	MS	F
Regression	1	0.867	0.867	0.197
Error	13	57.090	4.392
Total	14	57.957

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)

Answer 1

1.

x2 and x3 are the coefficients which concerns the departments.

So, the null and alternative hypothesis are,

H0:  B2 = B3 = 0

Ha: B2 /= 0 or B3 /= 0

2.

F = ((SSER - SSEF)/q ) / (SSEF / (n-k-1))

where SSER, SSEF are SS Error for reduced and full model.

q is number of coefficients in the null hypothesis

n is number of observations and k is number of predictors in full model

F = ((57.090 - 16.689)/2 ) / (16.689 / (15-3-1)) = 13.31449

3.

Degree of freedom = q, n-k-1 = 3, 15-3-1 = 3, 11

Critical value: Fα at α=0.05 and df = 3, 11 is   3.587

4.

Since the observed F statistic is greater than the critical value,

Reject H0.

There is evidence that the part of the model which concerns the departments is significant.