statistical inference, regression spss report
TRANSCRIPT
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Context:
The objective of the assignment is to study the regression analysis of the
validity co-efficient which we get in the correlation analysis of our sample.
For the analysis and learning purpose we had taken data from the BBA-16 (A &
B) of Graduate School of business at International Islamic University,
Islamabad. We recorded there CGPA, Intermediate percentage, medium of
instruction in Matric , intermediate institution, accounting1, accounting 2, cost
accounting, English1, English 2 and oral communication.
In the previous assignment we studied we the correlation and in this we will
study the regression of these variables.
We have made a comparison, of theirCGPA, Inter-Percentage, MOIM,
Public or Private Institution, Accounting Grades and Functional English
Grades in the Variable View by giving them appropriate values.
We study the following variables in our analysis:-
1) CGPA of the students, ( Dependent Variable)2) Intermediate Percentage, ( Independent Variable)3) Medium of Instruction up to Matric, ( Independent Variable)4) Grades of Accounting 1, 2 and 3, ( Independent Variable)5) Grades of Functional English 1 and 2, ( Independent Variable)6) Public or Private Sector, (Independent Variable).
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Regression Models:-
As we have 6 variables but 1 is dependent variable. So we has five validity co-
efficient; which are as given below
The above given validity co-efficient we get from Correlation Matrix.
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Question: Which is the Predictor we should study first?
The answer is that the validity co-efficient with highest value ofsignificance
will study first. The reason is because this has highest impact on the dependent
variable. So we should study these independent variable on there importance.
So we study Quantitative first as it has .526** level of significance.
After it we study Verbal which has .488** level of significance.
In the same we study MOIM, Inter-Percentage and Public orPrivate as they
have .261, .081 and .020 level of significance respectively.
So we have to study five Regression models.
Model # 1: =a+b1x1 (Quantitative)
Model # 2: =a+b2x2 (Verbal)
Model # 3: =a+b3x3 (MOIM)
Model # 4: =a+b4x4 (Inter-Percentage)
Model # 5: =a+b5x5 (Public or Private)
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DVs
IVs
CGPA
QUNT
Verbal
MOIM
Inter-pet
Istit
Dependent
Variable
Independent
Variables
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This is the simple model which we will study in this assignment.
At first;
When we open the data source file than this view will open, it has two views the
Data view and variable view these are explained in the previous assignment
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Model 1
(CGPA and QNT)
Scatter Dot
We want to calculate the regression, for this purpose we shall go in Graphs
then Legacy Dialogues and click on Scatter Dot. As shown in following
dialogue box
Theses are the steps for finding out the Scatter Dot Matrix;1. First go to Graphs2. Click on Legacy Dialogue3.Now click on Scatter Dot4. Select Simple Scatter5. Click Define6. Select CGPA on Y-axis.7. Select Quant on x-axis.8. Click Ok9. Simple Scatter is formed10.Double click on it.11.Click on Add to fit lines at Total.12.Click on Linear and now click on apply.13.A Linear curve is formed14.Click on Add to Fit Line.15.Click on Loess and then click on Apply.16.Loess Curve is formed.
These are the general steps that would be used to find out the regression of all
the variables.
In order to draw a Simple Scatter we have to follow following steps shown in
the fig.
These entire steps are those which are mention earlier
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When we click on scatter Dot, the following Dialogue box will appear.
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This is the close dialogue box of Scatter/ Dot; where we select the Simple
scatter
CGPA: is the dependent variable which we a studying
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Qunat: is the independent variable which relation on CGPA we are trying to
find out
Select the independent variables as well as dependent variable. In this case
independent variable is QNT and Dependent variable is CGPA.
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In this dialogue box, we shall click on Simple scatter Plot and then press OK.
Now the following dialogue box will appear.
Now we shall press OK. Now the graph will appear in the SPSS statistics
viewer as shown in the following diagram.
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The scatter plot appears in the out put view of the SPSS
Shown in the following diagram,
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We shall double click on the graph; the following dialogue box will appear.
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To ad fit lineinthe graph this is the procedure
This is the linear line which does not truly represent the data i.e. is does not
passes through maximum no of values. So is not the best method.
We can improve this by applying a much better method because our data does
not lies in linear symmetry.
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We go into the add fit line and select the LOESS curve this time,
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The Loess curve is shown in the graph;
It passes through the more data rather the linear curve
The loess curve suggests that instead of fitting a linear curve, we should fit a
quadratic curve.
So we will put the quadratic curve; procedure is explained in the figure.
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This graph shows the three curves .i.e. linear, loess and quadratic curve
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Now we will estimate that how quadratic curve is better from linear curve in
this situation.
Here we shall put the CGPA and QNT.
T
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heses are the steps
I. Go to analyzeII. Select the regression
III. Select linear regressionIV. Put the DV and IVV. Select the linear and quadratic
VI. Select Display ANOVAVII. Press OkThis is the procedure of finding out the estimation curve and in the next steps on
curves are shown rather the complete procedures
We also mark the option of Quadric and then mark display the anova table
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Now we shall press OK. Now the graph will appear in the SPSS statistics
viewer as shown in the following diagram.
Now we shall press OK. The graph will appear in the SPSS statistics viewer as
shown in the following diagram.
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Model 1
Model Description Table:
This table summarizes all the results of the model
Case Processing Company:
It tells us about total cases, excluded cases, forecasted cases, newly created
cases.
Variable Processing Summary:
It tells us about number of positive values, number of zeroes, number of
negative values, number of missing values of Dependent and Independent
Variables.
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Linear Tables:
CGPA of the student
Linear
Reporting Results
= a+ b1x1
=2.726+.150x1Interpretation
When the QNT is0 then CGPA is 2.726. per unit increase in QNT, CGPA
increase by .150
Intercept:
This is the predicted value of the response variable i.e. CGPA performance
when the predictor variables i.e. QNT is 0. In this case exam CGPA is 2.726,
when the QNT is 0.
The slope or the regression coefficient b1:
b1 is equal to .150 which is the change in the response variable i.e. CGPA when
the predictor variable i.QNT increases by one unit.
To simplify our interpretation, we can say that with an increase in QNT by 1
unit, the performance would increase by 0.150.
Model Summary
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CGPA
of the
student
R
Square
Adjusted R
Square
Std. Error of
the Estimate.283 .265 .304
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates of
Quadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the sales
during the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.
For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
The standard error in the modern summary suggests that we should fit a
quadratic model instead of a linear model.
For model comparison we can look at R2
only when both models have equal
number of predictors.
But in the current situation, in the linear model we have one predictor i.e QNT,
and in the second model we have two predictors QNT and QNT square. So in
this case instead of looking at the R square we shall be looking for model
comparison standard error is the best fit model.
The discussion of the quadratic model is beyond our syllabus.
ANOVA:
ANOVA
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Sum of
Squares df
Mean
Square F Sig.
Regressio
n
1.499 1 1.499 16.173 .000
Residual 3.800 41 .093
Total 5.299 42
The independent variable is QNT.
The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficient:
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
QNT .150 .037 .532 4.022 .000
(Constant
)
2.726 .156 17.467 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
Quadratic Tables:
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Quadratic
= a+ b1x1+b2x12
=3.310-1.86 x1+ .043x12
Model Summary:
Model Summary
R
R
Square
Adjusted R
Square
Std. Error of
the Estimate
.574 .330 .296 .298
The independent variable is QNT.
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates of
Quadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the salesduring the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.
For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
ANOVA:
ANOVA
Sum ofSquares df
MeanSquare F Sig.
Regressio
n
1.748 2 .874 9.846 .000
Residual 3.551 40 .089
Total 5.299 42
The independent variable is QNT.
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The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficients:
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
QNT -.186 .204 -.661 -.914 .366
QNT **
2
.043 .026 1.213 1.675 .102
(Constant
)
3.310 .381 8.697 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
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Here we have graph in which observed, linear and quadratic values are shown.
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Model 2
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Again the previous procedures are carried
This is the simple graph having linear curve
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This is final shape of the graph after having done the complete procedure to
apply the curves.
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Now the following dialogue box will appear and we shall interpret each
dialogue box.
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Model 2
(CGPA and Verbal)
Model Description Table:
This table summarizes all the results of the model
Model Description
Model Name MOD_1
Dependent
Variable
1 CGPA of the student
Equation 1 Linear
2 QuadraticIndependent Variable Verbal
Constant Included
Variable Whose Values Label
Observations in Plots
Unspecified
Tolerance for Entering Terms in
Equations
.0001
Case Processing Company:
It tells us about total cases, excluded cases, forecasted cases, newly created
cases
Case Processing
Summary
N
Total Cases 43
Excluded Casesa 1
Forecasted Cases 0
Newly Created
Cases
0
a. Cases with a missing
value in any variable are
excluded from the analysis.
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Variable Processing Summary:
It tells us about number of positive values, number of zeroes, number of
negative values, number of missing values of Dependent and Independent
Variables.
Variable Processing Summary
Variables
Dependent
Independe
nt
CGPA of
the student Verbal
Number of Positive Values 43 42
Number of Zeros 0 0
Number of Negative Values 0 0
Number of Missing
Values
User-Missing 0 0
System-Missing 0 1
Linear Tables:
CGPA of the student
Linear
Reporting Results
= a+ b2x2
=2.366+.206x2Interpretation
When the Verbal is0 then CGPA is 2.366. Per unit increase in Verbal, CGPA
increase by .206
Intercept:
This is the predicted value of the response variable i.e. CGPA when the
predictor variables i.e. Verbal is 0. In this case CGPA is 2.366, when the
Verbal is 0.
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The slope or the regression coefficient b1:
B2 is equal to .206 which is the change in the response variable i.e. CGPA when
the predictor variable i.Verbal increases by one unit.
To simplify our interpretation, we can say that with an increase in Verbal by 1
unit, the performance would increase by 0.206.
Model Summary
R
R
Square
Adjusted R
Square
Std. Error of
the Estimate
.440 .194 .174 .316
The independent variable is Verbal.
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates of
Quadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the sales
during the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.
For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
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ANOVA
Sum of
Squares df
Mean
Square F Sig.Regressio
n
.962 1 .962 9.625 .004
Residual 3.998 40 .100
Total 4.960 41
The independent variable is Verbal.
The ANOVA table shows us the overall impact of the model. It depicts theamount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficient:
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
Verbal .206 .066 .440 3.102 .004
(Constant
)
2.366 .318 7.451 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
There is no need to apply the quadratic eq to this model because standard error
is greater than the standard error of linear
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Model 3
If we carry out the previous procedures than this again will be the result of
model 3 where the variables are CGPA and Medium of instruction up to matric.
(CGPA and MOIM)
Curve Fit
[DataSet1] F:\Ijaz Bajwa. Data.sav
Warnings
The Quadratic model could not be fitted due to near-collinearity
among model terms.
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Model Description Table:
This table summarizes all the results of the model
Model Description
Model Name MOD_4
Dependent
Variable
1 CGPA of the student
Equation 1 Linear
Independent Variable Medium of Institution
upto Matric
Constant IncludedVariable Whose Values Label
Observations in Plots
Unspecified
Case Processing Company:
It tells us about total cases, excluded cases, forecasted cases, newly created
cases
Case Processing
Summary
N
Total Cases 43
Excluded Casesa
0
Forecasted Cases 0Newly Created
Cases
0
a. Cases with a missing
value in any variable are
excluded from the analysis.
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Variable Processing Summary:
It tells us about number of positive values, number of zeroes, number of
negative values, number of missing values of Dependent and Independent
Variables.
Variable Processing Summary
Variables
Dependent Independent
CGPA of
the student
Medium of
Institution
upto Matric
Number of Positive Values 43 27
Number of Zeros 0 16
Number of Negative Values 0 0
Number of Missing
Values
User-Missing 0 0
System-Missing 0 0
CGPA of the student
Linear
Reporting Results
= a+ b3x3
=3.218+.178x3
Interpretation
When the MOIM is 0 then CGPA is 3.218. Per unit increase in MOIM, CGPA
increase by .178
Intercept:
This is the predicted value of the response variable i.e. CGPA when the
predictor variables i.e. MOIM is 0. In this case CGPA is 3.218, when the
MOIM is 0.
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The slope or the regression coefficient b1:
B3 is equal to .178 which is the change in the response variable i.e. CGPA when
the predictor variable i.e MOIM increases by one unit.
To simplify our interpretation, we can say that with an increase in MOIM by 1
unit, the CGPA would increase by 0.178.
Model Summary
Model Summary
R
R
Square
Adjusted R
Square
Std. Error of
the Estimate
.237 .056 .033 .349
The independent variable is Medium of
Institution upto Matric.
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates ofQuadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the sales
during the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
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ANOVA
ANOVA
Sum ofSquares df
MeanSquare F Sig.
Regressio
n
.298 1 .298 2.440 .126
Residual 5.002 41 .122
Total 5.299 42
The independent variable is Medium of Institution upto Matric.
The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficient
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
Medium ofInstitution upto
Matric
.172 .110 .237 1.562 .126
(Constant) 3.218 .087 36.847 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
There is no need to apply the quadratic eq to this model because standard error
is greater than the standard error of linear.
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When we calculate quadratic the Sebecomes same so we dont need to apply the
quadratic eq.
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Model 4
(CGPA and Inter pct)
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Model Description Table:
This table summarizes all the results of the model
(CGPA and Inter pct)
Model Description
Model Name MOD_6
Dependent
Variable
1 CGPA of the student
Equation 1 Linear
2 Quadratic
Independent Variable Intermediate
Percentage of the
student
Constant Included
Variable Whose Values Label
Observations in Plots
Unspecified
Tolerance for Entering Terms in
Equations
.0001
Case Processing Company:
It tells us about total cases, excluded cases, forecasted cases, newly created
cases
Case Processing
Summary
N
Total Cases 43
Excluded Casesa 0
Forecasted Cases 0
Newly Created
Cases
0
a. Cases with a missing
value in any variable are
excluded from the analysis.
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Variable Processing Summary:
It tells us about number of positive values, number of zeroes, number of
negative values, number of missing values of Dependent and Independent
Variables.
Variable Processing Summary
Variables
Dependent Independent
CGPA of
the student
Intermediat
e
Percentage
of the
student
Number of Positive Values 43 43
Number of Zeros 0 0
Number of Negative Values 0 0
Number of MissingValues
User-Missing 0 0System-Missing 0 0
CGPA of the student
Linear
= a+ b4x4
=3.052+.004x4
Interpretation
When the inter pct is 0 then CGPA is 3.052. Per unit increase in inter pct,
CGPA increase by .178
Intercept:
This is the predicted value of the response variable i.e. CGPA when the
predictor variables i.e. Inter pct is 0. In this case CGPA is 3.0525, when the
inter pct is 0.
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The slope or the regression coefficient b1:
B4 is equal to .004 which is the change in the response variable i.e. CGPA when
the predictor variable i.e inter pct increases by one unit.
To simplify our interpretation, we can say that with an increase in inter pct by 1
unit, the CGPA would increase by 0.004
Model Summary
Model Summary
R
R
Square
Adjusted R
Square
Std. Error of
the Estimate.086 .007 -.017 .358
The independent variable is Intermediate
Percentage of the student.
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates of
Quadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the sales
during the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.
For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
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ANOVA
Sum of
Squares df
Mean
Square F Sig.Regressio
n
.040 1 .040 .309 .581
Residual 5.260 41 .128
Total 5.299 42
The independent variable is Intermediate Percentage of the
student.
The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficient
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
Intermediate
Percentage of the
student
.004 .007 .086 .556 .581
(Constant) 3.052 .496 6.151 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
There is no need to apply the quadratic eq to this model because standard error
is greater than the standard error of linear.
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There is no need to apply the quadratic eq to this model because standard error
is greater than the standard error of linear.
Model 5
(CGPA and Institution from which inter done)
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Model Description Table:
This table summarizes all the results of the model
(CGPA and Institution from which inter done)
Model Description
Model Name MOD_8
Dependent
Variable
1 CGPA of the student
Equation 1 Linear
2 QuadraticIndependent Variable Institution from which
interdone
Constant Included
Variable Whose Values Label
Observations in Plots
Unspecified
Tolerance for Entering Terms in
Equations
.0001
Case Processing Company:
It tells us about total cases, excluded cases, forecasted cases, newly created
cases
Case Processing
Summary
N
Total Cases 43
Excluded Casesa 0
Forecasted Cases 0
Newly Created
Cases
0
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Variable Processing Summary:
It tells us about number of positive values, number of zeroes, number of
negative values, number of missing values of Dependent and Independent
Variables.
Variable Processing Summary
Variables
Dependent Independent
CGPA of
the student
Institution
from which
interdone
Number of Positive Values 43 42
Number of Zeros 0 1
Number of Negative Values 0 0
Number of Missing
Values
User-Missing 0 0
System-Missing 0 0
CGPA of the student
Linear
= a+ b5x5
=3.379-.38x4
Interpretation
When the institution from which inter done is 0 then CGPA is 3.379. Per unit
increase in institution from which inter done, CGPA decrease by .38
Intercept:
This is the predicted value of the response variable i.e. CGPA when the
predictor variables i.e. institution from which inter done is 0. In this case
CGPA is 3.379, when the institution from which inter done is 0.
The slope or the regression coefficient b1:
B5 is equal to - .38 which is the change in the response variable i.e. CGPA when
the predictor variable i.e institution from which inter done increases by one unit.
To simplify our interpretation, we can say that with an increase in institution
from which inter done by 1 unit, the CGPA would decrease by 0.38
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Model Summary
Model Summary
RR
SquareAdjusted R
SquareStd. Error ofthe Estimate
.058 .003 -.021 .359
The independent variable is Institution from
which interdone.
It tells us R, R square, Adjusted R Square and standard error of the estimates. If
standard error of the estimates is less than the standard error of estimates of
Quadratic tables than we dont apply the quadratic eq and vice versa.
R square tells us how much variation in the dependent variable can be
accounted for by the independent variable.
Ris the sample correlation coefficient between the dependent variable (sales
during the year) and the independent variable.
Standard erroris measured in units of the response variable i.e. the sales
during the year and it tells us the standard distance of the data values from the
regression line or it tells us how far the values lie away from the regression line.
For different model comparisons, we always look at the standard error. The
model with smaller standard error will be a better model.
ANOVA
ANOVA
Sum of
Squares df
Mean
Square F Sig.
Regressio
n
.018 1 .018 .138 .712
Residual 5.282 41 .129
Total 5.299 42
The independent variable is Institution from which interdone.
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The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Coefficient
Coefficients
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.B Std. Error Beta
Institution from
which interdone
-.038 .102 -.058 -.371 .712
(Constant) 3.379 .154 21.907 .000
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
There is no need to apply the quadratic e.g. to this model because standard error
is greater than the standard error of linear.
There is no need to apply the quadratic e.g. to this model because standard error
is greater than the standard error of linear.
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Regression with more than one independent Variables
(= b0+ b1x1+b2x2+b3x3+b4 x4+b5 x5)
We shall go in analyze, click on regression then click on linear. When we click
on linear then following dialogue box will appear.
Now we shall select the dependent variable which is CGPA and independent
variable which are QNT, inter pct, Institution from which inter done, verbal,
MOIM.
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Now we shall click on statistics and check the following icons.
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When we click ok than following tables appear in out put file.
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Descriptive Statistics:
Regression
[DataSet1] E:\Education\IjazData\Statistical inference\IjazData.sav
Descriptive Statistics
Mean
Std.
Deviation N
CGPA of the student 3.3393 .34783 42
QNT 4.0119 1.27642 42
Medium of Institution
upto Matric
.64 .485 42
Intermediate
Percentage of the
student
67.02 7.592 42
Institution from which
interdone
1.43 .547 42
Verbal 4.7302 .74440 42
In this we find the mean and SD.
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Variables Entered/Removed:
Variables Entered/Removed
Model Variables Entered Variables Removed Method
1 Verbal, QNT,
Institution from
which interdone,
Intermediate
Percentage of the
student, Medium ofInstitution upto
Matrica
. Enter
a. All requested variables entered.
In this we can see which variables are entered and which are removed.
Model:We can input more than one model in a same regression command in SPSS.And in this column the numbers of models are shown.
Variables Entered:This column tells us about all the independent variable that we have specified
but did not blocked as SPSS allows us to enter variables in block for stepwise
regression.
Variables removed:Usually, this column is empty and only lists the removed variables when we do
stepwise regression.
Methods:The method used by SPSS to run regression is mentioned in this column.
Enter means that every independent variable was entered in usual manner.
But when stepwise regression is done, the entry tells us about that.
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Model Summary:
Model:This column tells us the number of models reported. As in SPSS we can input
more than one model in a same regression command.
R:
The correlation between the observed and predicted values of dependentvariable is called R and is the square root of R-Squared.
R-Square:R-Square is the proportion of variance in the dependent variable (CGPA) which
can be predicted from the independent variables (VER, QNT, Institution of
Inter, Inter-percentage and MOIM). This value indicates that 50.3% of the
variance in CGPA can be predicted from the variables VER, QNT, Institution
of Inter, Inter-percentage and MOIM. Note that this is an overall measure ofthe strength of association, and does not reflect the extent to which any
particular independent variable is associated with the dependent variable. R-
Square is also called the coefficient of determination.
Adjusted R-square:As predictors are added to the model, each predictor will explain some of the
variance in the dependent variable simply due to chance. One could continue to
add predictors to the model which would continue to improve the ability of the
Model Summary
Mod
el R
R
Square
Adjusted
R Square
Std. Error
of the
Estimate
Change Statistics
R Square
Change
F
Chang
e df1 df2
Sig. F
Change
1 .709a
.503 .434 .26171 .503 7.284 5 36 .000
a. Predictors: (Constant), Verbal, QNT, Institution from which interdone,
Intermediate Percentage of the student, Medium of Institution upto Matric
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predictors to explain the dependent variable, although some of this increase in
R-square would be simply due to chance variation in that particular sample. The
adjusted R-square attempts to yield a more honest value to estimate the R-
squared for the population. The value ofR-square was .503, while the value of
Adjusted R-square was .434, Adjusted R-squared is computed using the
formula 1 - ((1 - Rsq)(N - 1 )/ (N - k - 1)). From this formula, you can see that
when the number of observations is small and the number of predictors is large,
there will be a much greater difference between R-square and adjusted R-square
(because the ratio of (N - 1) / (N - k - 1) will be much greater than 1). By
contrast, when the number of observations is very large compared to the number
of predictors, the value of R-square and adjusted R-square will be much closer
because the ratio of (N - 1)/(N - k - 1) will approach 1.
Std. Error of the Estimate:The standard error of the estimate, also called the root mean square error, is the
standard deviation of the error term, and is the square root of the Mean Square
Residual (or Error).
ANOVA
ANOVA
Model
Sum of
Squares df
Mean
Square F Sig.
1 Regressio
n
2.495 5 .499 7.284 .000a
Residual 2.466 36 .068
Total 4.960 41
a. Predictors: (Constant), Verbal, QNT, Institution from which
interdone, Intermediate Percentage of the student, Medium of
Institution upto Matric
b. Dependent Variable: CGPA of the student
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The ANOVA table shows us the overall impact of the model. It depicts the
amount of variation in the response data explained by the predictor and the
amount of variation left unexplained.
We have the p-value which is the observed level of significance.
If p< , then we reject Ho (significant)
If p>, then we do not reject Ho (non-significant)
Model:This column tells us the number of models reported. As in SPSS we can input
more than one model in a same regression command.
This is the source of variance, Regression, Residual and Total. The Total
variance is partitioned into the variance which can be explained by theindependent variables (Regression) and the variance which is not explained by
the independent variables (Residual, sometimes called Error). Note that the
Sums of Squares for the Regression and Residual add up to the Total, reflecting
the fact that the Total is partitioned into Regression and Residual variance.
Sum of Squares:These are the Sum of Squares associated with the three sources of variance,
Total, Model and Residual. These can be computed in many ways.
Conceptually, these formulas can be expressed as:
SSTotal The total variability around the mean. S(Y - Ybar)2.
SSResidual The sum of squared errors in prediction. S(Y - Ypredicted)2.
SSRegression The improvement in prediction by using the predicted value
of Y over just using the mean of Y. Hence, this would be the squared
differences between the predicted value of Y and the mean of Y, S(Ypredicted -
Ybar)2. Another way to think of this is the SSRegression is SSTotal -
SSResidual. Note that the SSTotal = SSRegression + SSResidual.
Note that SSRegression / SSTotal is equal to 4.960, the value of R-Square.
This is because R-Square is the proportion of the variance explained by the
independent variables, hence can be computed by SSRegression / SSTotal.
Df:
These are the degrees of freedom associated with the sources of variance. Thetotal variance has N-1 degrees of freedom. In this case, there were N=43
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students, so the DF for total is 42. The model degrees of freedom corresponds
to the number of predictors minus 1 (K-1). You may think this would be 5-1
(since there were 5 independent variables in the model, VER, QNT, Institution
of Inter, Inter-percentage and MOIM). But, the intercept is automatically
included in the model (unless you explicitly omit the intercept). Including the
intercept, there are 6 predictors, so the model has 6-1=5 degrees of freedom.
The Residual degrees of freedom is the DF total minus the DF model, 42 - 5 is
37.
Mean Square:These are the Mean Squares; the Sum of Squares divided by their respective
DF. For the Regression, 2.495/ 5 = .499 . For the Residual, 2.466/ 36= .068 .These are computed so you can compute the F ratio, dividing the Mean Square
Regression by the Mean Square Residual to test the significance of the
predictors in the model.
F and Sig.:
The F-value is the Mean Square Regression (.499) divided by the Mean Square
Residual (0.068), yielding F=7.284. The p-value associated with this F valueis very small (0.000). These values are used to answer the question "Do the
independent variables reliably predict the dependent variable?". The p-value is
compared to your alpha level (typically 0.05) and, if smaller, you can conclude
"Yes, the independent variables reliably predict the dependent variable". You
could say that the group of variables VER, QNT, Institution of Inter, Inter-
percentage and MOIM can be used to reliably predict CGPA (the dependent
variable). If the p-value were greater than 0.05, you would say that the group of
independent variables does not show a statistically significant relationship withthe dependent variable, or that the group of independent variables does not
reliably predict the dependent variable. Note that this is an overall significance
test assessing whether the group of independent variables when used together
reliably predict the dependent variable, and does not address the ability of any
of the particular independent variables to predict the dependent variable. The
ability of each individual independent variable to predict the dependent variable
is addressed in the table below where each of the individual variables are listed.
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Coefficients:
It tells us whether the predictor variables have a significant effect on model or
not. If any predictor variable is not having significant impact then we exclude
that variable which has not significant impact on our linear eq.
Model:This column tells us the number of models reported. As in SPSS we can input
more than one model in a same regression command.
This column shows the predictor variables (constant, VER, QNT, Institution
of Inter, Inter-percentage and MOIM). The first variable (constant)
represents the constant, also referred to in textbooks as the Y intercept, the
Coefficients
a
Model
Unstandardized
Coefficients
Standardi
zed
Coefficie
nts
t Sig.
95.0% Confidence
Interval for B
B
Std.
Error Beta
Lower
Bound
Upper
Bound
1 (Constant) 1.623 .500 3.249 .003 .610 2.637
QNT .155 .036 .567 4.349 .000 .082 .227
Medium of
Institution upto
Matric
.094 .101 .131 .931 .358 -.111 .299
Intermediate
Percentage of the
student
.002 .006 .046 .348 .730 -.010 .014
Institution from
which interdone
.025 .080 .040 .312 .757 -.138 .188
Verbal .182 .063 .389 2.902 .006 .055 .309
a. Dependent Variable: CGPA of the student
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height of the regression line when it crosses the Y axis. In other words, this is
the predicted value ofCGPA when all other variables are 0.
B:These are the values for the regression equation for predicting the dependent
variable from the independent variable. These are called unstandardized
coefficients because they are measured in their natural units. As such, the
coefficients cannot be compared with one another to determine which one is
more influential in the model, because they can be measured on different
scales. For example, how can you compare the inter-percentage with the CGPA
scores? The regression equation can be presented in many different ways, for
example:
Y -Predicted = b0 + b1*x1 + b2*x2 + b3*x3 + b3*x3 +
b4*x4+b5*x5
The column of estimates (coefficients or parameter estimates, from here on
labeled coefficients) provides the values for b0, b1, b2, b3 and b4 for this
equation. Expressed in terms of the variables used in this example, the
regression equation is
CGPA = 1.623+ .155*QNT +.182*VER +.094*MOIM+.002*inter-
percent+.025* inter-institution
These estimates tell you about the relationship between the independent
variables and the dependent variable. These estimates tell the amount of
increase in science scores that would be predicted by a 1 unit increase in the
predictor. Note: For the independent variables which are not significant, the
coefficients are not significantly different from 0, which should be taken into
account when interpreting the coefficients. (See the columns with the t-value
and p-value about testing whether the coefficients are significant).
Std. Error:These are the standard errors associated with the coefficients. The standard
error is used for testing whether the parameter is significantly different from 0by dividing the parameter estimate by the standard error to obtain a t-value (see
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the column with t-values and p-values). The standard errors can also be used to
form a confidence interval for the parameter, as shown in the last two columns
of this table.
Beta:These are the standardized coefficients. These are the coefficients that youwould obtain if you standardized all of the variables in the regression, including
the dependent and all of the independent variables, and ran the regression. By
standardizing the variables before running the regression, you have put all of the
variables on the same scale, and you can compare the magnitude of the
coefficients to see which one has more of an effect. You will also notice that
the larger betas are associated with the larger t-values.
T and Sig.:These columns provide the t-value and 2 tailed p-value used in testing the null
hypothesis that the coefficient/parameter is 0. If you use a 2 tailed test, then
you would compare each p-value to your preselected value of alpha.
Coefficients having p-values less than alpha are statistically significant. For
example, if you chose alpha to be 0.05, coefficients having a p-value of 0.05 or
less would be statistically significant (i.e., you can reject the null hypothesis and
say that the coefficient is significantly different from 0). If you use a 1 tailedtest (i.e., you predict that the parameter will go in a particular direction), then
you can divide the p-value by 2 before comparing it to your preselected alpha
level. However, if you used a 2-tailed test and alpha of 0.01, the p-value of
.0255 is greater than 0.01 and the coefficient for variable would not be
significant at the 0.01 level. Had you predicted that this coefficient would be
positive (i.e., a one tail test), you would be able to divide the p-value by 2
before comparing it to alpha. This would yield a one-tailed p-value of 0.00945,
which is less than 0.01 and then you could conclude that this coefficient isgreater than 0 with a one tailed alpha of 0.01.
The constant is significantly different from 0 at the 0.05 alpha levels.
However, having a significant intercept is seldom interesting.
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The coefficient forQNT (.155) is statistically significantly different from 0
using alpha of 0.05 because its p-value is 0.000, which is smaller than 0.05.
The coefficient forMOIM(.094) is not statistically significantly different
from 0 because its p-value is definitely larger than 0.05.The coefficient forVER(.182) is statistically significant because its p-value
of 0.008 is less than .05.
The coefficient forINTER.Inst (.025) is not significantly different form 0
because its p-value is greatly larger than 0.05.
The coefficient forINTER-Percent (.002) is not significantly different from
0 because its value is definitely larger than 0.05.
Reporting Results (Combine)= b0+ b1x1+b2x2+b3x3+b4 x4+b5 x5
=1.623+.155 x1+ .182x2+.094x3+ .002x4+ .025x5
Interpretation:
When all the independent variables are 0 then the CGPA is 1.623. Per unitincrease in QNT, the value of CGPA increase by .155; Per unit increase in
verbal, CGPA increase by .182; Per unit increase in MOIM, CGPA increase by
.094; Per unit increase in inter pct, CGPA increase by .002; Per unit increase in
institute from which inter done, CGPA increase by .025.
Intercept:
This is the predicted value of the response variable i.e. CGPA when the
predictor variables is 0. In this case CGPA is 1.623, when the all the predictorsare 0.
The slope or the regression coefficient:
When all the independent variables are 0 then the CGPA is 1.623. Per unit
increase in QNT, the value of CGPA increase by .155; Per unit increase in
verbal, CGPA increase by .182; Per unit increase in MOIM, CGPA increase by
.094; Per unit increase in inter pct, CGPA increase by .002; Per unit increase in
institute from which inter done, CGPA increase by .025.
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