Chapter 14 Simple Linear Regression
Chapter 14 Simple Linear
Regression
Outline
The Simple Linear Regression Model
Regression Model the Regression Equations Estimated Regression Equation
The Least Squares Method The Coefficient of Determination
Correlation coefficient
Measures of Association Between Two Variables
Chapter 3 section 5 Covariance Correlation Coefficient
1
Covariance ? conceptual formula
Sample Covariance
Sample variance
Positive values indicate a positive relationship.
Negative values indicate a negative relationship.
Correlation Coefficient
Sample Correlation Coefficient
The coefficient can take on values between -1 and +1:
Values near -1 indicate
Values near 0 indicate
Values near +1 indicate
a strong negative
no linear relationship.
a strong positive linear
linear relationship.
relationship.
rxy= -1
rxy= 0
rxy= +1
Equations for calculation
Sample covariance applied equation
2
Algebra 1 Review
Draw a line between two points (x2,y2)
General line equation
where, m is the slope
(x1,y1) Intercept b
b is the intercept
(you need the slope and one point)
Simple Linear Regression
Draw a line among many points General line equation
where, b1 is the estimated slope
Intercept b
b0 is the intercept
(you need the slope and one point)
Example: Armand's Pizza
xi
yi
250
Quarterly Pizza Sales
200
150
100
50
0
0
10
20
30
14
130
Student Population
3
Example: Armand's Pizza
x
y
Pop Sales
(000) ($000)
2 58
6 105
8 88
8 118
12 117
16 137
20 157
20 169
22 149
26 202
140 1300
X2 4 36 64 64
144 256 400 400 484 676 2528
y2 xy 3364 116 11025 630 7744 704 13924 944 13689 1404 18769 2192 24649 3140 28561 3380 22201 3278 40804 5252 184730 21040
Determine the slope = 5
Example: Armand's Pizza
x
y
Pop Sales
(000) ($000)
2 58
6 105
8 88
8 118
12 117
16 137
20 157
20 169
22 149
26 202
140 1300
X2 4 36 64 64
144 256 400 400 484 676 2528
y2 xy
3364 11025
116 630
Determine the intercept
7744 704
13924 944 13689 1404
= 60
18769 24649
2192 3140
Write the regression equation
28561 3380
22201 3278
40804 5252
184730 21040
What is the estimated quarterly pizza sales for a store with 8 thousand students on campus?
with 15 thousand students?
Quarterly Pizza Sales
250
200
150 135 100
50
0
0
10 15 20
30
Student Population
4
Quarterly Pizza Sales
250
200
150
100
50
0
0
10
20
30
Student Population
Quarterly Pizza Sales
250
200
150
100
50
0
0
10
20
30
Student Population
8 thousand students
88 100 118
5
20 thousand students
149 160 169
Seven Steps of Hypothesis Testing
Where:
1.
You believe there is a positive relationship
between X and Y
Where:
1.
You believe there is a negative relationship
between X and Y
4.
0
4.
0
5. Use Excel
5. Use Excel
6
EXCEL Regression Output
6. .0000255 < .01,Reject Ho, Accept Ha The sign of the slope must correspond to the alternate hypothesis
7. Student population is significant.
7
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