stat modelling assignment 5

7/26/2019 Stat Modelling Assignment 5

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STQS 2234

STATISTICAL MODELLING

TITLE

ASSIGNMENT 5

PREPARED FOR

DR. MARINA BINTI ZAHARI

GROUP 7

NORAINSYIRAH BINTI MOHAMED NORDIN A151588

NUR SYAHIDAH BINTI KHALAPIAH A150105

NUR NADIRAH BINTI MOHAMAD YOHYI A151055

SITI NUR ZAWANIE BINTI MD SOBRI A149121

NUR AZILA BINTI BAHARUDDIN A148328

NURFARIHA NADHIRAH BINTI AHMAD A151675

NOORMARINA BINTI MOKHTAR A151110


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1. Consider the multiple linear regression model y = X + . If denotes the least squares

estimator of , show that + [ 1 ] .

y = X +

Minimizes SSE; SSE = 2=1 where ; = = ( )( ) = + =

= 0

Derived from fitted model ;

1

From multiple linear regression model y = X + ;

y = X +

= X +

[ 1 ][ + ] + [ 1 ]

QUESTION 1:


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3(e)

i. Residual versus x1 -

ii. Residual versus x2

iii. Residual versus x3iv. Residual versus x4


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QUESTION 2:


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QUESTION 3

a) 0:0 j H ,

:1 H At least one of the j is not equal to zero

5,...,2,1 j

Test Statistic: 81.40 f

Compare 81.40 f > 17.419,5,01.0 f and p-value = 0.0052 < 01.0

Here, we reject 0 H . We conclude that the data is linearly related to 4321 ,,, x x x x and 5 x .

0:

0:

11

10

H

H

Test Statistic: 47.20 t

Compare 093.247.2 19,2/05.00 t t and p-value = 0.023 < 05.0

With 05.0 , we would reject the null hypothesis. This indicates the predictor contribute tothe model.

0:

0:

21

20

H

H





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0:

0:

31

30

H

H




0:

0:

41

40

H

H




0:

0:

51

50

H

H


Compare 093.225.0 19,2/05.00 t t and p-value = 0.801 > 05.0

With 05.0 , we fail to reject the null hypothesis. This indicates the predictor could bedeleted from the model.

b) When 5 x was excluded from the model and the model was re-fitted, it shows that the model is

better compare to the model with 5 x .

c) Some of the residuals are having large number and these become the outliers.


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d) In (a), 2adj R is the least compared to the2

adj R in (b) and (c).2

adj R for (b) is higher than (a) and

lower than (c) when 5 x is removed with 25 observations. For (c), the2

adj R is the highest when

5 x is removed with 24 observations. It shows that the2

adj R increases as test. This is because the

variables in the model are all useful for the model.

e)

Residuals versus 1 x

- Based on the plots, the points are randomly scattered. More points are plotted at the bottom of the graph or at negative region. This is because it is over-predicted. There arealso some outliers.


- Based on the plots, the points are randomly scattered. More points are plotted at the bottom of the graph or at negative region. This is because it is over-predicted. There arealso some outliers. Points at the left bottom are plotted closely to each other.


- Based on the plots, the points are randomly scattered. More points are plotted at the bottom of the graph or at negative region. This is because it is over-predicted. There arealso some outliers. The points at the negative region of the graph are mostly at the same xvalue.


- Based on the plots, the points are randomly scattered. The points plotted are most likelysame at both regions. There are also some outliers.


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f) 0:0 j H ,

:1 H At least one of the j is not equal to zero

4,3,2,1 j

Test Statistic: 79.210 f

Compare 79.210 f > 87.220,4,05.0 f

Here, we reject 0 H . We conclude that the data is linearly related to 321 ,, x x x and 4 x .

0:

0:

11

10

H

H


Compare 725.176.5 20,2/05.00 t t


0:

0:

21

20

H

H


Compare 725.196.5 20,2/05.00 t t



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0:

0:

31

30

H

H


Compare 725.190.2 20,2/05.00 t t


0:

0:

41

40

H

H


Compare 725.199.4 20,2/05.00 t t


g) The residual plots against 321 ,, x x x and 4 x has boundary between -1 and 1 and there are no

pattern shown in the plots. All of the plots have an outlier. The points in the all plots aresymmetrically distributed and most of the points are near to zero.


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QUESTION 4 :

Based on the 2 -value criterion, the best model is the model with the two predictors PctComp

and PctTD as the 2-value give a substantial increase by jumps from 64.8 to 85.1.

Based on the adjusted 2 -value and MSE criteria, the best model is the model with the seven

predictors Att, PctComp, Yds, YdsperAtt, TD, PctTD and PctInt as the model have the largest

adjusted 2 -value (100.0) and the smallest (5.1).

Based on the criterion, there are eight possible best models

i. the model with 6 predictors containing Att, PctComp, Yds, YdsperAtt, PctTD and

PctInt;

ii. the model with 6 predictors containing Att, Comp, PctComp, YdsperAtt, PctTD and

PctInt;

iii. the model with 7 predictors containing Att, PctComp, Yds, YdsperAtt, TD, PctTD and

PctInt;

iv. the model with 7 predictors containing Att, PctComp, Yds, YdsperAtt, PctTD, Int and

PctInt;

v. the model with 8 predictors containing Att, PctComp, Yds, YdsperAtt, TD, PctTD, Int

and PctInt;

vi. the model with 8 predictors containing Att, Comp, PctComp, Yds, YdsperAtt, TD,

PctTD and PctInt;

vii. the model with 9 predictors containing Att, PctComp, Yds, YdsperAtt, TD, PctTD, Lng,

Int and PctInt;

viii. and the model with 9 predictors containing Att, Comp, PctComp, Yds, YdsperAtt, TD,

PctTD, Int and PctInt.

As all of those models are unbiased models, because their values equal (or are below) the

number of parameters, .


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QUESTION 5:

stat modelling assignment 5

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