SIMPLE LINEAR REGRESSION
Regression – it is a test used to possibly approximate or predict
value of one variable in terms of the other variable.
Equation of the Regression Line:
ypred = mx + b where ypred = the predicted value of y
b= Y – mX
m = [n∑(xy) – ∑x ∑y] / [n∑x2 –(∑x)2]
Example:
Below are the data of 10 out of school youth who took the
placement examination offered by the Department of Education
(DepEd) to those school drop outs who desire to go back to
school. The data show the number of school years the students
had attended and the corresponding placement examination
percentile rank each student had attained.
Solution:
Students
No. of Yrs.
Percentile Rank
x^2
xy
x
y
1
4
58
16
232
2
5
79
25
395
3
5
73
25
365
4
3
59
9
177
5
6
90
36
540
6
4
64
16
256
7
5
76
25
380
8
6
85
36
510
9
4
76
16
304
10
3
49
9
147
TOTAL
45
709
213
3306
X=
∑x / n = 45/10 = 4.5
Y = ∑y /n = 709/10 = 70.9
m=
[n∑(xy) – ∑x ∑y] / [n∑x2 –(∑x)2]
=[10(3306) – 45 (709)] / [10(213) – 452)
=
m=
(33060 – 31905) / (2130 – 2025)
11
b=
Y – mX
= 70.9 – 11(4.5)
b = 21.4
Therefore the equation of the regression line for the given
data is:
ypred = mx + b = 11x + 21.4
*** Use the equation of the regression line to check whether
the visually approximated value of y = 44 when x = 2 is accurate
or nearly accurate.
ypred = 11x + 21.4 = 11 (2) + 21.4 = 43.4
*** What is the approximate percentile rank of a student who
attended 3 ½ years in high school?
ypred = 11x + 21.4 = 11 (3.5) + 21.4 = 59.9
STANDARD EROR OF ESTIMATE (Se)
it is a measure of the amount of spread of the sample points
about the regression line
is shows how the sample points deviate from their regression
line.
Se = √( ∑y2 – b ∑y – m ∑xy) / (n-2)
Example:
Compute the Se for the set of paired data of the 10 out-of-school
youth who took the DepEd placement examination.
Student
1
2
3
4
5
6
7
8
9
10
Total
No. of Yrs.
x
4
5
5
3
6
4
5
6
4
3
45
Percentile Rank
Y
58
79
73
59
90
64
76
85
76
49
709
y2
xy
3364
6241
5329
3481
8100
4096
5776
7225
5776
2401
51789
232
395
365
177
540
256
380
510
304
147
3306
Se =
=
√( ∑y2 – b ∑y – m ∑xy) / (n-2)
√ [51789 – 21.4 (709) – 11(3,306)] / (10-2)
=
=
=
√ (51789 – 15, 192.6 – 36,366) / 8
√ 31.3
5.59