Coefficient of Corr and Deter

8/13/2019 Coefficient of Corr and Deter

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The McGraw-Hill Companies, Inc., 2000

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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11-8


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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11-9


70605040

150

140

130

120

Age

Pressure

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150

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


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


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


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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11-25

11-4 Example

70605040

150

140

130

120

Age

Pressure

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150

140

130

120

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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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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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).

Coefficient of Corr and Deter

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