© the mcgraw-hill companies, inc., 2000 correlationandregression further mathematics - core
TRANSCRIPT
© The McGraw-Hill Companies, Inc., 2000
Correlation
and
Regression
Further Mathematics - CORE
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Outline
11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression
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Outline
11-5 Coefficient of
Determination and
Standard Error of
Estimate
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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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Objectives
Find the coefficient of determination.
Find the standard error of estimate.
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11-2 Scatter Plots
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.
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11-2 Scatter Plots - Example
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.
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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
Pre
ssur
e
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Age
Pre
ssur
ePositive Relationship
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11-2 Scatter Plots - Other Examples
15105
90
80
70
60
50
40
Number of absences
Fin
al g
rade
15105
90
80
70
60
50
40
Number of absences
Fin
al g
rade
Negative Relationship
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11-2 Scatter Plots - Other Examples
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10
5
0
X
Y
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5
0
x
yNo Relationship
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11-3 Correlation Coefficient
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,
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11-1311-3 Range of Values for the
Correlation Coefficient
Strong negativerelationship
Strong positiverelationship
No linearrelationship
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11-1411-3 Formula for the Correlation
Coefficient r
r
n xy x y
n x x n y y
2 2 2 2
Where n is 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
.443 112 ,399 20
634 47= ,819= ,34522
r
givesrforformulatheinngSubstituti
yx
xyyx
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11-1611-3 The Significance of the
Correlation Coefficient
The population correlation coefficient, , is the correlation between 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
rwith d f n
2
12
2
. .
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11-19 11-3 Example
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
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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).
11-3 Example
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11-21
Step 4: Make the decision. Reject the null hypothesis, since the test value falls 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 y intercept 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, a represents the intercept and b
the slope of the line.
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11-25 11-4 Example
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Age
Pre
ssur
e
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Age
Pre
ssur
e
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 dependent variable (y) for a given value of the independent variable (x).
Caution: Use x values within the experimental region when predicting y values.
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11-27 11-4 Example
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) = 129.248 129.2
Note that the value of 50 is within the range of x values.
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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 of the variation of the dependent variable that is explained by the regression line and the independent variable.
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11-2911-5 Coefficient of Determination and Standard Error of Estimate
r2 is the square of the correlation coefficient.
The coefficient of nondetermination is (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 standard deviation of the observed y values 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
nor
sy 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 + 8.13x and n = 6, find sest.
Here, a = 55.57, b = 8.13, and n = 6. Substituting into the formula gives sest
= 6.48 (verify).