1. 24/02/11
Tutorial 3
Inferential Statistics, Statistical Modelling & Survey Methods
(BS2506)
Pairach Piboonrungroj (Champ)

2. 1. House price (Again) Analysis of Variance (ANOVA) Predictor
24/02/11
1. House price (Again)
Predictor
(Variable)
Coefficient (B)
SE (B)
Constant
-2.5
41.4 X1
1.62
0.21 X2
0.257
1.88 X4
-0.027
0.008
Analysis of Variance (ANOVA)
Source of variation
Sum of Squares
Degree of Freedom
Mean Squares
Regression
277,895
Residual
34,727

3. 1 (a) (i) Write out the estimated regression equation Predictor
24/02/11
1 (a)
(i) Write out the estimated regression equation
Predictor
(Variable)
Coefficient (B)
SE (B)
Constant
-2.5
41.4 X1
1.62
0.21 X2
0.257
1.88 X4
-0.027
0.008

4. 1 (a) Step1: Critical Value Step2: t-Statistic At 1%
24/02/11
1 (a)
(ii) Test for the significance of regression equation
At 1%
Step1: Critical Value
Step2: t-Statistic 4

5. 24/02/11
1 (a)
(ii) Test for the significance of regression equation
Step1: Critical Value
At 1%
Step2: t-Statistic
>
Reject H0
Do NOT
Reject H0
<
<
Reject H0 5

6. 1. a). (iii) What are DF for SSR & SSE?
24/02/11
1. a). (iii) What are DF for SSR & SSE?
Predictor
(Variable)
Coefficient (B)
SE (B)
Constant
-2.5
41.4 X1
1.62
0.21 X2
0.257
1.88 X4
-0.027
0.008
Analysis of Variance (ANOVA)
Source of variation
Sum of Squares
Degree of Freedom
Mean Squares
Regression
277,895
3 (p)
Residual
34,727
11 (n-p-1) 6

7. Analysis of Variance (ANOVA)
24/02/11
1. a). (iv) Test for Significant relationship X&Y?
H0:
H1: At least one of the coefficients does not equal 0
Analysis of Variance (ANOVA)
Source of variation
Sum of Squares
Degree of Freedom
Mean Squares
F Statistic
Regression
277,895 3
92,631
29.341
Residual
34,727 11
3157
Critical Value At
Then we can reject Null hypothesis, there is a relationship between Xs & Y 7

8. Analysis of Variance (ANOVA)
24/02/11
1. a). (v) Compute the coefficient of determination and explain its meaning R2
Analysis of Variance (ANOVA)
Source of variation
Sum of Squares
Degree of Freedom
Mean Squares
F Statistic
Regression
277,895 3
92,631
29.341
Residual
34,727 11
3157
TOTAL
312,622
R2 = 1 – (34,727/312,622)
R2 = 1 – 0.111
R2 = = 88.9% 8

9. 24/02/11
1(b)
Model 1
Model 2
Model 3

10. 24/02/11
1(b)
(i) Compute Adjusted Coefficient of determination for three models 10

11. 24/02/11
1(b)
(ii) Interpret the coefficients on the house type, Beta5 and Beta6
(model 2)
Prices for Detached houses increase by £63,794
Prices for Terrace Houses decreased by £65,371
(relative to Semi- detached) 11

12. 24/02/11
1(b)
(iii) At 0.05 level of significance, determine whether model 2 is superior to model1
Model 1
Model 2
Significant i.e., Model 2 is better than Model 1 12

13. 24/02/11
1(b)
(iv) At 0.05 level of significance, determine whether model 3 is superior to model 2
Model 2
Model 3
NOT Significant i.e., Model 3 is NOT better than Model 2 13

14. 1(b)
(v) From model2, estimate the price of 5 years old detached house with 250 square meters

15. 2. Advertising expenditure
24/02/11
2. Advertising expenditure
X, Advertising
(£000)
Y, Sales
5.5 90
2.0 40
3.2 55
6.0 95
3.8 70
4.4 80
5.0
6.5
7.0 88 85 92 91
R square
0.97
Adjusted R Square
0.96
Standard error of regression
3.37
Analysis of variance DF
Sum Square
Mean Square
Regression
2,904
Residual
80.0
Variables in the Equation
Variable B
SE B
Advert
31.79
4.48
Advert-square
-2.30
0.485
(constant)
-17.22
9.65 15

16. 2.(a) State the regression equation for the curvilinear model.
Variables in the Equation
Variable B
SE B
Advert
31.79
4.48
Advert-square
-2.30
0.485
(constant)
-17.22
9.65

17. 2.(b) Predict the monthly sales (in pounds) for a month with total advertising expenditure of £6,000

18. 2.(c) Determine there is significant relationship between the sales and advertising expenditure at the 0.01 level of significance
H0:
H1: At least one of the coefficients does not equal 0
Analysis of variance DF
Sum Square
Mean Square F
Regression 2
2,904
1,452
127.05
Residual 7
80.0
11.428
Critical Value At
Then we can reject Null hypothesis, there is a curvilinear relationship between sales and advertising expenditure

19. 2 (d) Fit a linear model to the data and calculate SSE for this model

20. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales 1
5.5 90 2 40 3
3.2 55 4 6 95 5
3.8 70
4.4 80 7 88 8 85 9
6.5 92 10 91

21. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales xy
x^2
y^2 1
5.5 90
495
30.25
8100 2 40 80 4
1600 3
3.2 55
176
10.24
3025 6 95
570 36
9025 5
3.8 70
266
14.44
4900
4.4
352
19.36
6400 7 88
528
7744 8 85
425 25
7225 9
6.5 92
598
42.25
8464 10 91
637 49
8281
Sum
49.4
786
4127
266.54
64764
Average
4.94
78.6
412.7
26.654
6476.4

22. 2 (d) Fit a linear model to the data and calculate SSE for this model

23. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales xy
x^2
y^2 1
5.5 90
495
30.25
8100 2 40 80 4
1600 3
3.2 55
176
10.24
3025 6 95
570 36
9025 5
3.8 70
266
14.44
4900
4.4
352
19.36
6400 7 88
528
7744 8 85
425 25
7225 9
6.5 92
598
42.25
8464 10 91
637 49
8281
Sum
49.4
786
4127
266.54
64764
Average
4.94
78.6
412.7
26.654
6476.4

24. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales xy
x^2
y^2
predicted 1
5.5 90
495
30.25
8100
84.68 2 40 80 4
1600
46.70 3
3.2 55
176
10.24
3025
59.72 6 95
570 36
9025
90.10 5
3.8 70
266
14.44
4900
66.23
4.4
352
19.36
6400
72.74 7 88
528
7744 8 85
425 25
7225
79.25 9
6.5 92
598
42.25
8464
95.53 10 91
637 49
8281
100.95
Sum
49.4
786
4127
266.54
64764
Average
4.94
78.6
412.7
26.654
6476.4

25. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales xy
x^2
y^2
predicted
Square
Error 1
5.5 90
495
30.25
8100
84.68
28.35 2 40 80 4
1600
46.70
44.92 3
3.2 55
176
10.24
3025
59.72
22.29 6 95
570 36
9025
90.10
24.00 5
3.8 70
266
14.44
4900
66.23
14.20
4.4
352
19.36
6400
72.74
52.69 7 88
528
7744
4.41 8 85
425 25
7225
79.25
33.05 9
6.5 92
598
42.25
8464
95.53
12.43 10 91
637 49
8281
100.95
99.01
Sum
49.4
786
4127
266.54
64764
Average
4.94
78.6
412.7
26.654
6476.4

26. 2 (d) Fit a linear model to the data and calculate SSE for this modelID X
Advertising Y
Sales xy
x^2
y^2
predicted
Square
Error 1
5.5 90
495
30.25
8100
84.68
28.35 2 40 80 4
1600
46.70
44.92 3
3.2 55
176
10.24
3025
59.72
22.29 6 95
570 36
9025
90.10
24.00 5
3.8 70
266
14.44
4900
66.23
14.20
4.4
352
19.36
6400
72.74
52.69 7 88
528
7744
4.41 8 85
425 25
7225
79.25
33.05 9
6.5 92
598
42.25
8464
95.53
12.43 10 91
637 49
8281
100.95
99.01
Sum
49.4
786
4127
266.54
64764
335.36
Average
4.94
78.6
412.7
26.654
6476.4

27. 24/02/11
2(e) At 0.01 level of significance, determine whether the curvilinear model is superior to the linear regression model
Curvilinear Model
Linear Regression Model
Significant i.e., Curvilinear effect make significant contribution and should be included in the model. 27

28. 2 (f) Draw a scatter diagram between the sales& Advertising expenditure.

29. 2 (f) Sketch the Linear regression

30. 2 (f) Sketch the Quadratic regression
Linear Regression

31. Thank you Download this Slides at www.pairach.com/teaching
Q & A