Correlation and Regression

1
Chapter 11
11-1

• Correlation and Regression

2
Outline
11-2

• 11-1 Introduction
• 11-2 Scatter Plots
• 11-3 Correlation
• 11-4 Regression

3
Outline
11-3

• 11-5 Coefficient of Determination and
  Standard Error of Estimate

4
Objectives
11-4

• Draw a scatter plot for a set of ordered pairs.
• Find the correlation coefficient.
• Test the hypothesis H0 ? 0.
• Find the equation of the regression line.

5
Objectives
11-5

• Find the coefficient of determination.
• Find the standard error of estimate.
• Find a prediction interval.

6
11-2 Scatter Plots
11-6

• A scatter plot is a graph of the ordered pairs
  (x, y) of numbers consisting of the independent
  variable, x, and the dependent variable, y.

7
11-2 Scatter Plots - Example
11-7

• Construct a scatter plot for the data obtained in
  a study of age and systolic blood pressure of six
  randomly selected subjects.
• The data is given on the next slide.

8
11-2 Scatter Plots - Example
11-8
S
u
b
e
c
t
A
e
,

x
P
r
e
s
s
u
r
e

y
,
g
j
A
4
3
1
2
8
B
4
8
1
2
0
C
5
6
1
3
5
D
6
1
1
4
3
E
6
7
1
4
1
F
7
0
1
5
2
9
11-2 Scatter Plots - Example
11-9
Positive Relationship
e
1
5
0
e
r
1
5
0
u
r
u
s
s
s
s
e
e
r
r
P
1
4
0
P
1
4
0
1
3
0
1
3
0
1
2
0
1
2
0
7
0
6
0
5
0
4
0
7
0
6
0
5
0
4
0
A
g
e
A
g
e
10
11-2 Scatter Plots - Other Examples
11-10
Negative Relationship
11
11-2 Scatter Plots - Other Examples
11-11
No Relationship
12
11-3 Correlation Coefficient
11-12

• The correlation coefficient computed from the
  sample data measures the strength and direction
  of a relationship between two variables.
• Sample correlation coefficient, r.
• Population correlation coefficient, ??

13
11-3 Range of Values for the Correlation
Coefficient
11-13
Strong negative relationship
Strong positive relationship
No linear relationship
??
??
?
14
11-3 Formula for the Correlation Coefficient r
11-14
(
)
(
)
(
)
?
?
?
?
n
xy
x
y
?
r
(
(
(
(
)
)
)
)
?
?
?
?
?
?
?
?
?
?
2
2
n
x
x
n
y
y
2
2
Where n is the number of data pairs
15
11-3 Correlation Coefficient - Example (Verify)
11-15

• Compute the correlation coefficient for the age
  and blood pressure data.

x
y
x
y
3
4
5
8
1
9
4
7
6
3
4
,


,


,

?
?
?
x
y
2
0
3
9
9
1
1
2
4
4
3
,
,
,
.



2
2
?
?
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r
0
8
9
7
.
.



















16
11-3 The Significance of the Correlation
Coefficient
11-16

• The population corelation coefficient, ?, is the
  correlation between all possible pairs of data
  values (x, y) taken from a population.

17
11-3 The Significance of the Correlation
Coefficient
11-17

• H0 ?? 0 H1 ??? 0
• This tests for a significant correlation between
  the variables in the population.

18
11-3 Formula for the t tests for the
Correlation Coefficient
11-18
?
n
2
?
t
?
r
1
2
?
?
with
d
f
n
2

.
.
19
11-3 Example
11-19

• Test the significance of the correlation
  coefficient for the age and blood pressure data.
  Use ? 0.05 and r 0.897.
• Step 1 State the hypotheses.
• H0 ?? 0 H1 ??? 0

20
11-3 Example
11-20

• Step 2 Find the critical values. Since ?
  0.05 and there are 6 2 4 degrees of freedom,
  the critical values are t 2.776
  and t 2.776.
• Step 3 Compute the test value. t
  4.059 (verify).

21
11-3 Example
11-21

• Step 4 Make the decision. Reject the null
  hypothesis, since the test value falls in the
  critical region (4.059 gt 2.776).
• Step 5 Summarize the results. There is a
  significant relationship between the variables of
  age and blood pressure.

22
11-4 Regression
11-22

• The scatter plot for the age and blood pressure
  data displays a linear pattern.
• We can model this relationship with a straight
  line.
• This regression line is called the line of best
  fit or the regression line.
• The equation of the line is y a bx.

23
11-4 Formulas for the Regression Line y
a bx.
11-23
(
)
(
(
(
)
)
)
?
?
?
?
?
y
x
x
xy
2
?
?
a

(
(
)
)
?
?
?
2
n
x
x
2
(
)
(
)
(
)
?
?
?
?
n
xy
x
y
?
b

(
)
(
)
?
?
?
2
n
x
x
2
Where a is the y intercept and b is the slope
of the line.
24
11-4 Example
11-24

• Find the equation of the regression line for the
  age and the blood pressure data.
• Substituting into the formulas give a
  81.048 and b 0.964 (verify).
• Hence, y 81.048 0.964x.
• Note, a represents the intercept and b the slope
  of the line.

25
11-4 Example
11-25
y ? 81.048 0.964x
26
11-4 Using the Regression Line to
Predict
11-26

• The regression line can be used to predict a
  value for the dependent variable (y) for a given
  value of the independent variable (x).
• Caution Use x values within the experimental
  region when predicting y values.

27
11-4 Example
11-27

• Use the equation of the regression line to
  predict the blood pressure for a person who is 50
  years old.
• Since y 81.048 0.964x, theny 81.048
  0.964(50) 129.248?? 129.
• Note that the value of 50 is within the range of
  x values.?

28
11-5 Coefficient of Determination and Standard
Error of Estimate
11-28

• The coefficient of determination, denoted by r2,
  is a measure of the variation of the dependent
  variable that is explained by the regression line
  and the independent variable.

29
11-5 Coefficient of Determination and Standard
Error of Estimate
11-29

• r2 is the square of the correlation coefficient.
• The coefficient of nondetermination is (1 r2).
• Example If r 0.90, then r2 0.81.

30
11-5 Coefficient of Determination and Standard
Error of Estimate
11-30

• The standard error of estimate, denoted by sest,
  is the standard deviation of the observed y
  values about the predicted y values.
• The formula is given on the next slide.

31
11-5 Formula for the Standard Error of
Estimate
11-31
(
)
?
2
?
y
y

?
s
?
n
2
est
or
?
?
?
?
?
y
a
y
b
xy
2
?
s
?
n
2
est
32
11-5 Standard Error of Estimate - Example
11-32

• From the regression equation, y
  55.57 8.13x and n 6, find sest.
• Here, a 55.57, b 8.13, and n 6.
• Substituting into the formula gives sest
  6.48 (verify).

33
11-5 Prediction Interval
11-33

• A prediction interval is an interval constructed
  about a predicted y value, y , for a specified
  x value.

34
11-5 Prediction Interval
11-34

• For given ? value, we can state with (1 ?)100
  confidence that the interval will contain the
  actual mean of the y values that correspond to
  the given value of x.

35
11-5 Formula for the Prediction Interval about a
Value y
11-35
2
-
)
(
1
X
x
n



-
1
s
t
y
2
est
a
2
2
(
)
n
å
-
å
x
x
n
2
-
)
(
1
X
x
n




1
s
t
y
2
est
a
2
2
(
)
n
å
-
å
x
x
n
?
?
2
.
.
n
f
d
with
36
11-5 Prediction interval - Example
11-36

• A researcher collects the data shown on the next
  slide and determines that there is a significant
  relationship between the age of a copy machine
  and its monthly maintenance cost. The regression
  equation is y 55.57 8.13x. Find the 95
  prediction interval for the monthly maintenance
  cost of a machine that is 3 years old.

37
11-5 Prediction Interval - Example
11-37
A
1
62
B
2
78
C
3
70
D
4
90
E
4
93
F
6
103
38
11-5 Prediction Interval - Example
11-38

• Step 1 Find ?x, ?x2 and . ?x 20, ?x2
  82,
• Step 2 Find y ? for x 3. y 55.57
  8.13(3) 79.96
• Step 3 Find sest sest 6.48 as shown in
  previous example.

39
11-5 Prediction Interval - Example
11-39

• Step 4 Substitute in the formula and solve.
  t?/2 2.776, d.f. 6 2 4 for 95 60.53 lt
  y lt 99.39 (verify)Hence, one can be 95
  confident that the interval 60.53 lt y lt 99.39
  contains the actual value of y.