# Appendix 5

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APPENDIX 5                                                                                                            175

ASSUMPTIONS OF CORRECTNESS OF REGRESSION MODEL

The Indian models for both Tech marks and English at Tech need to be
diagnosed for assumption correctness.

Technikon Mark

Checking the assumption that normality of residuals holds true for both
factors.

Histogram
Dependent Variable: TECHMARK
12

10

8

6

4
Frequency

Std. Dev = .97
2
Mean = 0.00
0                                                                                     N = 41.00
-2.00                     -1.00          0.00         1.00          2.00
-1.50                 -.50          .50          1.50          2.50

Regression Standardized Residual

Normal P-P Plot of Regression Standardized Residual
Dependent Variable: TECHMARK
1.00

.75
Expected Cum Prob

.50

.25

0.00
0.00          .25          .50       .75          1.00

Observed Cum Prob

As can be seen from both the histogram of residuals and the normal
probability plot the residuals from the above model seem to have a normal
distribution.
This is further substantiated by performing a Kolmogorov-Smirnov and a
Shapiro-Wilk Test for normality, the results of which are highlighted below.
APPENDIX 5                                                                                                           176

Tests of Normality
a
Kolmogorov-Smirnov                         Shapiro-Wilk
Statistic      df        Sig.         Statistic        df        Sig.
Standardized Residual                                  .076         41       .200*           .976           41       .620
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction

The p-value is 0.62 for the Shapiro and 0.020 for the Kolmogorov tests,
indicating there is not sufficient evidence to reject the possibility of normality of
residuals in the model.

Further linearity diagnostics on the model by running the standard residual
and dependent variable plots : i.e :

a)                  Standardized Residuals vrs Predicted Values.
b)                  Dependent Variable vrs the first independent Variable.
c)                  Dependent variable vrs the second independent Variable

(a)
3

2

1
Standardized Residual

0

-1

-2
0               10               20                   30

Unstandardized Predicted Value
APPENDIX 5                                                                   177

(b)
Partial Regression Plot
Dependent Variable: TECHMARK
10

0

-10
TECHMARK

-20
-60      -40         -20       0        20   40

ENGTECH

(c )

Partial Regression Plot
Dependent Variable: TECHMARK
10

8

6

4

2

0
TECHMARK

-2

-4

-6
-20               -10         0       10        20

SHOLMARK

There does not seem to be any significant patterns of assumption violation
from any of the three diagrams for the model.
APPENDIX 5                                                                                                                                                                 178

Technikon English Mark

Checking the assumption of normality of residuals holds true for both factors.
Histogram
Dependent Variable: ENGTECH
12

10

8

6

4
Frequency

Std. Dev = .98
2
Mean = .02

0                                                                                                N = 42.00
-4.00                       -3.00       -2.00     -1.00          0.00         1.00
-3.50                           -2.50   -1.50         -.50          .50          1.50

Regression Standardized Residual

Normal P-P Plot of Regression Standardized Residual
Dependent Variable: ENGTECH
1.00

.75

.50
Expected Cum Prob

.25

0.00
0.00            .25          .50              .75            1.00

Observed Cum Prob

While the Kolmogorov test provided an acceptable result (R2 = 0.20) the
Shapiro-Wilk test did not (R2 = 0.10). As can be seen from both the histogram
of residuals and the normal probability plot the residuals from the above
model does not seem to have a normal distribution.

Tests of Normality
a
Kolmogorov-Smirnov                                            Shapiro-Wilk
Statistic      df        Sig.                            Statistic       df         Sig.
Standardized Residual                                                                    .103         42       .200*                              .893           42       .010**
*. This is a lower bound of the true significance.
**. This is an upper bound of the true significance.
a. Lilliefors Significance Correction
APPENDIX 5                                                                      179

In the Shapiro -Wilk test, the p-value is 0.01 indicating there is sufficient
evidence to reject the possibility of normality of residuals in the model.

Further diagnostics on the Shapiro model by running the standard residual
plots : i.e :

a) Standardized Residuals vrs Predicted Values.
b) Dependent variables vrs the first independent Variable.
c) Dependent vrs the second independent Variable

indicate outcomes as indicated below.

(a)

2

1

0

-1
Standardized Residual

-2

-3

-4
20       30       40        50    60   70   80

Unstandardized Predicted Value
APPENDIX 5                                                                  180

(b)
Partial Regression Plot
Dependent Variable: ENGTECH
30

20

10

0

-10

-20
ENGTECH

-30

-40
-10                       0               10

TECHMARK

(c)

Partial Regression Plot
Dependent Variable: ENGTECH
20

10

0

-10

-20
ENGTECH

-30

-40
-3    -2    -1    0       1   2   3        4

ENGTECH1

The above model violates the assumptions of normality of residuals, this is
possibly caused by the existence of a large outlier. The outlier was found in
record 9, which was deleted. This resulted in the model below. As can be
seen, R 2 has improved to 0.828 from 0.768 . Also the model assumptions in
this case are not violated.
APPENDIX 5                                                                                                                                       181

Model Summary

Std. Error
Model                           R      R Square                                 R Square                     Estimate
1                                .895a     .801                                     .796                       6.3117
2                                .910b     .828                                     .819                       5.9480
a. Predictors: (Constant), TECHMARK
b. Predictors: (Constant), TECHMARK, ENGTECH1

a
Coefficients

Standardi
zed
Unstandardized                                      Coefficien
Coefficients                                          ts
Model                                                         B        Std. Error                                  Beta         t      Sig.
1                       (Constant)                           21.920        3.273                                               6.697     .000
TECHMARK                              2.276         .184                                      .895    12.376     .000
2                       (Constant)                           21.526        3.089                                               6.969     .000
TECHMARK                              1.904         .232                                      .749     8.203     .000
ENGTECH1                              1.821         .757                                      .220     2.406     .021
a. Dependent Variable: ENGTECH

Histogram
Dependent Variable: ENGTECH
6

5

4

3

2
Frequency

Std. Dev = .97
1
Mean = .02
0                                                                                       N = 41.00
-1.75   -1.25    -.75          -.25          .25         .75          1.25
-1.50   -1.00       -.50          0.00         .50         1.00          1.50

Regression Standardized Residual
APPENDIX 5                                                                                                         182

Normal P-P Plot of Regression Standardized Residual
Dependent Variable: ENGTECH
1.00

.75

Expected Cum Prob
.50

.25

0.00
0.00        .25       .50      .75        1.00

Observed Cum Prob

Tests of Normality
a
Kolmogorov-Smirnov                        Shapiro-Wilk
Statistic      df        Sig.        Statistic        df        Sig.
Standardized Residual                                 .074         41       .200*          .951           41       .115
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction

The p value is 0.115 indicating there is not sufficient evidence to reject the
possibility of normality of residuals in the model.

Further linearity diagnostics on the model were taken by running the standard
residual and dependent variable plots : i.e :

a) Standardized Residuals vrs Predicted Values.
b) Dependent variables vrs the first independent Variable.
c) Dependent vrs the second independent Variable
APPENDIX 5                                                                                          183

(a)
2.0

1.5

1.0

.5

0.0
Standardized Residual

-.5

-1.0

-1.5

-2.0
20            30        40           50   60       70        80

Unstandardized Predicted Value

(b)

Partial Regression Plot
Dependent Variable: ENGTECH
30

20

10

0

-10
ENGTECH

-20

-30
-10                                   0                      10

TECHMARK

(c )

Partial Regression Plot
Dependent Variable: ENGTECH
20

10

0
ENGTECH

-10

-20
-3        -2        -1        0        1   2    3        4

ENGTECH1

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