Chapter 10 Section 2.ppt

Section 10.2-1
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

Section 10.2-2
Chapter 10
Correlation and Regression
10-1 Review and Preview
10-2 Correlation
10-3 Regression
10-4 Prediction Intervals and Variation
10-5 Multiple Regression
10-6 Nonlinear Regression

Section 10.2-3
Key Concept
In part 1 of this section introduces the linear correlation
coefficient, r, which is a number that measures how well
paired sample data fit a straight-line pattern when
graphed.
Using paired sample data (sometimes called bivariate
data), we find the value of r (usually using technology),
then we use that value to conclude that there is (or is not)
a linear correlation between the two variables.

Section 10.2-4
Key Concept
In this section we consider only linear relationships,
which means that when graphed, the points approximate
a straight-line pattern.
In Part 2, we discuss methods of hypothesis testing for
correlation.

Section 10.2-5
Part 1: Basic Concepts of Correlation

Section 10.2-6
Definition
A correlation exists between two variables when the
values of one are somehow associated with the values
of the other in some way.
A linear correlation exists between two variables when
there is a correlation and the plotted points of paired
data result in a pattern that can be approximated by a
straight line.

Section 10.2-7
Exploring the Data
We can often see a relationship between two variables by
constructing a scatterplot.
The following slides show scatterplots with different characteristics.

Section 10.2-8
Scatterplots of Paired Data

Section 10.2-9
Scatterplots of Paired Data

Section 10.2-10
Requirements for
Linear Correlation
1. The sample of paired (x, y) data is a simple random
sample of quantitative data.
2. Visual examination of the scatterplot must confirm that
the points approximate a straight-line pattern.
3. The outliers must be removed if they are known to be
errors. The effects of any other outliers should be
considered by calculating r with and without the
outliers included.

Section 10.2-11
Notation for the
Linear Correlation Coefficient
2
number of pairs of sample data
denotes the addition of the items indicated
sum of all -values
indicates that each -value should be squared and then those squares add
n
x x
x x



 
2
ed
indicates that each -value should be added and the total then squared
indicates each -value is multiplied by its corresponding -value. Then sum those up.
linear correlation
x x
xy x y
r


coefficient for sample data
linear correlation coefficient for a population of paired data


Section 10.2-12
The linear correlation coefficient r measures the strength
of a linear relationship between the paired values in a
sample. Here are two formulas:
Technology can (and should) compute this.
Formula
r =
n Sxy
( )- (Sx)(Sy)
n(Sx2
) - (Sx)2
n(Sy2
) - (Sy)2
 
1
x y
z z
r
n




Section 10.2-13
Interpreting r
Using Table A-6: If the absolute value of the computed
value of r, exceeds the value in Table A-6, conclude that
there is a linear correlation. Otherwise, there is not
sufficient evidence to support the conclusion of a linear
correlation.
Using Software: If the computed P-value is less than or
equal to the significance level, conclude that there is a
linear correlation. Otherwise, there is not sufficient
evidence to support the conclusion of a linear correlation.

Section 10.2-14
Caution
Know that the methods of this section apply to a linear
correlation.
If you conclude that there does not appear to be linear
correlation, know that it is possible that there might be
some other association that is not linear.

Section 10.2-15
Properties of the
Linear Correlation Coefficient r
1. – 1 ≤ r ≤ 1
2. If all values of either variable are converted to a different
scale, the value of r does not change.
3. The value of r is not affected by the choice of x and y.
Interchange all x- and y-values and the value of r will not
change.
4. r measures strength of a linear relationship.
5. r is very sensitive to outliers, which can dramatically
affect the value of r.

Section 10.2-16
Example
The paired shoe / height data from five males are listed
below. Use a computer or a calculator to find the value
of the correlation coefficient r.

Section 10.2-17
Example - Continued
Requirement Check: The data are a simple random
sample of quantitative data, the plotted points appear to
roughly approximate a straight-line pattern, and there
are no outliers.

Section 10.2-18
Example - Continued
A few technologies are displayed below, used to
calculate the value of r.

Section 10.2-19
Using the Formulas to
Calculate Correlation
Technology is highly recommended, and as such, we
refer you to the textbook, pages 501 and 502 for the
manual calculations using the formulas.

Section 10.2-20
Is There a Linear Correlation?
We found previously for the shoe and height example that
r = 0.591.
We now proceed to interpret its meaning.
Our goal is to decide whether or not there appears to be a
linear correlation between shoe print lengths and heights
of people.
We can base our interpretation on a P-value or a critical
value from Table A-6.

Section 10.2-21
Interpreting the Linear
Correlation Coefficient r
Using computer software:
If the P-value is less than the level of significance,
conclude there is a linear correlation.
Our example with technologies provided a P-value of
0.294.
Because that P-value is not less than the significance
level of 0.05, we conclude there is not sufficient evidence
to support the conclusion that there is a linear correlation
between shoe print length and heights of people.

Section 10.2-22
Interpreting the Linear
Correlation Coefficient r
Using Table A-6:
Table A-6 yields r = 0.878 for five pairs of data and a 0.05
level of significance. Since our correlation was r = 0.591,
we conclude there is not sufficient evidence to support the
claim of a linear correlation.

Section 10.2-23
Interpreting r:
Explained Variation
The value of r2 is the proportion of the variation in y that is
explained by the linear relationship between x and y.

Section 10.2-24
Example
r = 0.591.
With r = 0.591, we get r2 = 0.349.
We conclude that about 34.9% of the variation in height
can be explained by the linear relationship between
lengths of shoe prints and heights.

Section 10.2-25
Common Errors
Involving Correlation
1. Causation: It is wrong to conclude that correlation
implies causality.
2. Averages: Averages suppress individual variation and
may inflate the correlation coefficient.
3. Linearity: There may be some relationship between x
and y even when there is no linear correlation.

Section 10.2-26
Caution
Know that correlation does not imply causality.

Section 10.2-27
Part 2: Formal Hypothesis Test

Section 10.2-28
Formal Hypothesis Test
We wish to determine whether there is a significant linear
correlation between two variables.
Notation:
n = number of pairs of sample data
r = linear correlation coefficient for a sample of paired data
ρ = linear correlation coefficient for a population of paired data

Section 10.2-29
Hypothesis Test for Correlation
Requirements
1. The sample of paired (x, y) data is a simple random
sample of quantitative data.
2. Visual examination of the scatterplot must confirm that
the points approximate a straight-line pattern.
3. The outliers must be removed if they are known to be
errors. The effects of any other outliers should be
considered by calculating r with and without the
outliers included.

Section 10.2-30
Hypotheses
Critical Values: Refer to Table A-6.
P-values: Refer to technology.
Test Statistic: r
0
1
: 0 (There is no linear correlation.)
: 0 (There is a linear correlation.)
H
H





Section 10.2-31
If | r | > critical value from Table A-6, reject the null
hypothesis and conclude that there is sufficient evidence
to support the claim of a linear correlation.
If | r | ≤ critical value from Table A-6, fail to reject the null
hypothesis and conclude that there is not sufficient
evidence to support the claim of a linear correlation.

Section 10.2-32
Example
r = 0.591.
Conduct a formal hypothesis test of the claim that there is
a linear correlation between the two variables.
Use a 0.05 significance level.

Section 10.2-33
Example - Continued
We test the claim:
With the test statistic r = 0.591 from the earlier example.
The critical values of r = ± 0.878 are found in Table A-6
with n = 5 and α = 0.05.
We fail to reject the null and conclude there is not
sufficient evidence to support the claim of a linear
correlation.
0
1
: 0 (There is no linear correlation)
: 0 (There is a linear correlation)
H
H





Section 10.2-34
P-Value Method for a Hypothesis
Test for Linear Correlation
2
1
2
r
t
r
n



The test statistic is below, use n – 2 degrees of freedom.
P-values can be found using software or Table A-3.

Section 10.2-35
Example
Continuing the same example, we calculate the test
statistic:
Table A-3 shows this test statistic yields a P-value that is
greater than 0.20. Technology provides the P-value as
0.2937.
2 2
0.591
1.269
1 1 0.591
2 5 2
r
t
r
n
  
 
 

Section 10.2-36
Example - Continued
Because the P-value of 0.2937 is greater than the
significance level of 0.05, we fail to reject the null
hypothesis.
We conclude there is not sufficient evidence to support
the claim of a linear correlation between shoe print length
and heights.

Section 10.2-37
One-Tailed Tests
One-tailed tests can occur with a claim of a positive linear
correlation or a claim of a negative linear correlation. In
such cases, the hypotheses will be as shown here.
For these one-tailed tests, the P-value method can be used as in
earlier chapters.

Chapter 10 Section 2.ppt

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Chapter 10 Section 2.ppt