coefficient of corr and deter
TRANSCRIPT
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11-1
Correlation and
Regression
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11-2
Outline
11-1 Introduction
11-2 Scatter Plots
11-3 Correlation
11-4 Regression
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11-3
Outline
11-5 Coefficient of
Determination and
Standard Error of Estimate
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11-4
Objectives
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.
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11-5
Objectives
Find the coefficient of
determination.
Find the standard error of
estimate.
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11-6
11-2 Scatter Plots
Ascatter plotis a graph of the
ordered pairs (x , y)of numbersconsisting of the independent
variable, x, and the dependent
variable, y.
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11-7
11-2 Scatter Plots - Example
Construct a scatter plot for the data
obtained in a study of age and systolicblood pressure of six randomly selected
subjects.
The data is given on the next slide.
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11-2 Scatter Plots - Example
Subject Age, x Pressure, y
A 43 128
B 48 120
C 56 135
D 61 143
E 67 141
F 70 152
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11-2 Scatter Plots - Example
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Age
Pressure
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140
130
120
Age
Pressure
Positive Relationship
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11-1011-2 Scatter Plots - Other Examples
15105
90
80
70
60
50
40
Number of absences
Finalgrade
15105
90
80
70
60
50
40
Number of absences
Finalgrade
Negative Relationship
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11-1111-2 Scatter Plots - Other Examples
706050403020100
10
5
0
X
Y
706050403020100
10
5
0
x
y
No Relationship
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11-12
11-3 Correlation Coefficient
The correlation coefficient
computed from the sample datameasures the strength and direction
of a relationship between two
variables. Sample correlation coefficient, r.
Population correlation coefficient,
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11-1311-3 Range of Values for the
Correlation Coefficient
Strong negative
relationship
Strong positive
relationship
No linear
relationship
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11-1411-3 Formula for the Correlation
Coefficient r
rn xy x y
n x x n y y
22 2 2
Where nis the number of data pairs
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11-1511-3 Correlation Coefficient -
Example (Verify)
Compute the correlation coefficient
for the age and blood pressure data.
.897.0
.443112,39920
63447=,819=,345
22
r
givesrforformulatheinngSubstituti
yx
xyyx
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11-1611-3 The Significance of the
Correlation Coefficient
The population correlation
coefficient, , is the correlationbetween all possible pairs of
data values (x , y) taken from a
population.
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11-1711-3 The Significance of the
Correlation Coefficient
H0: = 0 H1: 0 This tests for a significant
correlation between the variables
in the population.
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11-1811-3 Formula for the t-tests for the
Correlation Coefficient
tn
r
with d f n
2
1
2
2
. .
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11-19
11-3 Example
Test the significance of the correlation
coefficient for the age and bloodpressure data. Use = 0.05 andr= 0.897.
Step 1:State the hypotheses.
H0: = 0 H1: 0
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11-20
Step 2:Find the critical values. Since
= 0.05 and there are 6
2 = 4 degreesof freedom, the critical values are
t= +2.776 and t= 2.776.
Step 3: Compute the test value.
t= 4.059 (verify).
11-3 Example
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11-21
Step 4:Make the decision. Reject the
null hypothesis, since the test valuefalls in the critical region (4.059 > 2.776).
Step 5: Summarize the results. There is
a significant relationship between the
variables of age and blood pressure.
11-3 Example
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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.
11-4 Regression
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11-2311-4 Formulas for the Regression
Line y = a+ bx.
ay x x xy
n x x
bn xy x y
n x x
2
2 2
2 2
Where a is the yintercept and b is
the slope of the line.
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11-24
11-4 Example
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, arepresents the intercept and b
the slopeof the line.
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11-4 Example
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Age
Pressure
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Age
Pressure
y
= 81.048 + 0.964x
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11-2611-4 Using the Regression Line to
Predict
The regression line can be used to
predict a value for the dependentvariable (y) for a given value of the
independent variable (x).
Caution: Use xvalues within theexperimental region when
predicting yvalues.
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11-2811-5 Coefficient of Determination
and Standard Error of Estimate
The coefficient of determination,
denoted by r2, is a measure ofthe variation of the dependent
variable that is explained by the
regression line and theindependent variable.
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11-2911-5 Coefficient of Determination
and Standard Error of Estimate
r2is the square of the correlation
coefficient. The coefficient of
nondeterminationis (1 r2).
Example:If r= 0.90, then
r2= 0.81.
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11-3011-5 Coefficient of Determination
and Standard Error of Estimate
The standard error of estimate,
denoted by sest, is the standarddeviation of the observed yvalues about the predicted y
values. The formula is given on the next
slide.
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11-3111-5 Formula for the Standard
Error of Estimate
s
y y
n
or
s y a y b xy
n
est
est
2
2
2
2
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11-3211-5 Standard Error of Estimate -
Example
From the regression equation,
y = 55.57 + 8.13xand n= 6, find sest. Here, a= 55.57, b= 8.13, and n= 6.
Substituting into the formula gives
sest= 6.48 (verify).