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SUBJECT: BASIC AND INFERENTIAL STATISTICS
REPORTER:SHIELA ROBETH B. VINARAO
TOPIC: REGRESSION ANALYSIS
PROFESSOR: DR. GLORIA T. MIANO
REGRESSION
ANALYSIS
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THE SIMPLE LINEAR
REGESSION ANALYSIS
The simple linear regression
analysis is used when there is a
signifcant relationshipbetween and variables.
This is used in predicting thevalue o a dependent variable
given the value o the
independent variable .
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THE SIMPLE LINEAR
REGESSION ANALYSIS
Suppose the advertising cost
and sales are correlated, then
we can predict the uture sales
in terms o advertising cost .Another type o problem which
uses regression analysis is when
variables corresponding to yearsare given, it is possible to
predict the value o that variable
several years hence or several
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THE SIMPLE LINEAR
REGESSION ANALYSIS
F
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M
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LA
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ExampleConsider the following data:
1 6
2 4
3 3
4 5
5 4
6 2
1 6
2 4
3 3
4 5
5 4
6 2
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
x-axis
y-axis
Straight line indicates that the two variables are to some
extent LINEARLY RELATED
The variable we are basing our predictions on is called the
predictor variable and is referred to as . When there is only
one predictor variable, the prediction method is called
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1 6 6 1
2 4 8 4
3 3 9 9
4 5 20 16
5 4 20 25
6 2 12 36
3.5
4
= 4
75
1 6 6 1
2 4 8 4
3 3 9 9
4 5 20 16
5 4 20 25
6 2 12 36
SIMPLE LINEAR REGESSION
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3.5
4
= 4
75
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05Example A study isconducted on the
relationship of the
number of absences
and the grades of
the students in
English.
Determine the
relationship using
the following data.
•
Number of Absences Grades in English
1 90
2 852 80
3 75
3 80
8 65
6 70
1 95
4 80
5 80
5 75
1 92
2 89
1 80
9 65
1 90
2 85
2 80
3 75
3 80
8 65
6 70
1 95
4 80
5 80
5 75
1 92
2 89
1 80
9 65
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Number of
Absences
Grades in
English
1 90
2 85
2 80
3 75
3 80
8 65
6 70
1 95
4 80
5 805 75
1 92
2 89
1 80
9 65
1 90
2 85
2 80
3 75
3 80
8 65
6 70
1 95
4 80
5 805 75
1 92
2 89
1 80
9 65
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
Number of absences
x
ra!es in "n#lis$
y
Scatter Diagram
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Solving by the Stepwise Method
Problem :Is there a significant relationship betweenthe number of absences and the grades of15 students in English class?
Hypotheses :
Level of significance :
There is no significant relationship between the
number of absences and the grades of 15 students
in English class. Ho:
There is a significant relationship between the
number of absences and the grades of 15 students
in Englishclass.
H1:
df = n – 2
= 15 – 2
= 13
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ST
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S
Pearson Product Moment
Coefficient of Correlation
281 7335 3950
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The computedr value of is beyond the critical value
of at level of significance with degrees of freedom, so the
null hypothesis is rejected.
This means that there is a significant relationship
between the number of absences and the grades of
students in English. Since the value ofr is negative, it
implies that students who had more absences had lower
grades.
Decision Rule :
If ther computed value is greater than or beyondthe critical value, reject Ho.
Conclusion :
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Suppose we want to predict the grade of the student
who has incurred7 absences. To get the value of x, the
simple linear regression analysis will be used.
69 is the grade of
the student with
7 absences.
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Remarks: It is important to remember that thevalues of a and b are only estimates of the
corresponding parameters of a and b.
To justify the assumption of linearity, atest for linearity of regression should be
performed.
If there are two or more independent
variables, the regression equation becomes
+
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The significance ofthe slope ofthe regression line is to determine if
the regression model is usable.
If theslope is not equal to zero, then we can usethe regression
model to predict the dependent variable for any value of the
independent variable.
If theslope is equal to zero, we do not usethe model to makepredictions.
The scatter plot amounts to determining whether or not the slope
of the line of the best fit is significantly different from a horizontal
line or not.
A horizontal line means there is no association between two
variables, that is
In testing for significance in simple linear regression, the null
hypothesis is H0: and the alternative hypothesis is H1:
Significance Test in Simple
Linear Regression
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0%5 1 1%5 2 2%5 3 3%5 4 4%5 5 5%50
0%2
0%4
0%6
0%8
1
1%2
x-axis
y-axis
If
A horizontal line means there is no
association between two variables.
Slope of
Linear Regression
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Significance Test in Simple
Linear Regression
The t-test is conducted for testing the significance of r todetermine if the relationship is not a zero correlation.
Where:
FO
RM
U
L
A
1
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Significance Test in Simple
Linear Regression
The t-test is conducted for testing the significance of r todetermine if the relationship is not a zero correlation.
Where:
is the estimated standard deviation of
is the standard deviation of the values about
the regression line.
FO
RM
U
L
A
2
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ExampleGiven are two sets of data on the number of customers(in hundreds) and sales (in thousand of pesos) for a given
period of time from ten eateries. Find the equation of the
regression line which can predict the amount of sales
from the number of customers. Can we conclude that wecan use the model to make such a prediction?
Eatery 1 2 3 4 5 6 7 8 9 10
x2 6 8 8 12 16 20 20 22 26
y 58 105 88 118 117 137 157 169 149 202
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2 58 116 4 3364
6 105 630 36 11025
8 88 704 64 7744
8 118 944 64 13924
12 117 1404 144 13689
16 137 2192 256 18769
20 157 3140 400 24649
20 169 3380 400 28561
22 149 3278 484 22201
26 202 5252 676 40804
2 58 116 4 3364
6 105 630 36 11025
8 88 704 64 7744
8 118 944 64 13924
12 117 1404 144 13689
16 137 2192 256 18769
20 157 3140 400 24649
20 169 3380 400 28561
22 149 3278 484 22201
26 202 5252 676 40804
Significance Test in Simple
Linear Regression
Given:
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2 58 116 4 3364 70 144 144
6 105 630 36 11025 90 64 225
8 88 704 64 7744 100 36 144
8 118 944 64 13924 100 36 324
12 117 1404 144 13689 120 4 9
16 137 2192 256 18769 140 4 9
20 157 3140 400 24649 160 36 9
20 169 3380 400 28561 160 36 81
22 149 3278 484 22201 170 64 441
26 202 5252 676 40804 190 144 144
2 58 116 4 3364 70 144 144
6 105 630 36 11025 90 64 225
8 88 704 64 7744 100 36 144
8 118 944 64 13924 100 36 324
12 117 1404 144 13689 120 4 9
16 137 2192 256 18769 140 4 9
20 157 3140 400 24649 160 36 9
20 169 3380 400 28561 160 36 81
22 149 3278 484 22201 170 64 441
26 202 5252 676 40804 190 144 144
Significance Test in Simple
Linear Regression
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Since8.62 is greater than 3.355, wereject the H0 or accept the H1. Thus,
the obtained relationship is
significant or is non zero using .005level.
We can conclude that we can usethe model to predict sales from
population.
Decision:
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COMPUTATION USING
MICROSOFT EXCEL
Pearsonr
Slopeb
Intercepta
Syntax
+pearson(array1,array2)
or+correl(array1,array2)
+slope(known_y’s,known_x’s)
+intercept(known_y’s,known_x’s)
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PEARSONr
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SLOPEb INTERCEPTa
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THANK YOU