Chapter 10 Section 3.ppt

Section 10.3-1
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

Section 10.3-2
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 10.3-3
Key Concept
In Part 1 of this section we find the equation of the straight
line that best fits the paired sample data. That equation
algebraically describes the relationship between two
variables.
The best-fitting straight line is called a regression line and
its equation is called the regression equation.
In Part 2, we discuss marginal change, influential points,
and residual plots as tools for analyzing correlation and
regression results.

Section 10.3-4
Part 1: Basic Concepts of Regression

Section 10.3-5
Regression
The typical equation of a straight line y = mx + b is
expressed in the form ŷ = b0 + b1x, where b0 is the y-
intercept and b1 is the slope.
The regression equation expresses a relationship
between x (called the explanatory variable, predictor
variable or independent variable), and ŷ (called the
response variable or dependent variable).

Section 10.3-6
Definitions
Regression Equation:
Given a collection of paired sample data, the regression
line (or line of best fit, or least-squares line) is the
straight line that “best” fits the scatterplot of data.
The regression equation ŷ = b0 + b1x algebraically
describes the regression line.

Section 10.3-7
Notation for
Regression Equation
Population
Parameter
Sample
Statistic
y-Intercept of
regression equation
β0 b0
Slope of regression
equation
β1 b1
Equation of the
regression line
y = β0 + β1x ŷ = b0 + b1x

Section 10.3-8
Requirements
1. The sample of paired (x, y) data is a random sample
of quantitative data.
2. Visual examination of the scatterplot shows that the
points approximate a straight-line pattern.
3. Any outliers must be removed if they are known to be
errors. Consider the effects of any outliers that are
not known errors.

Section 10.3-9
Formulas for b1 and b0
Technology will compute these values.
1
Slope: y
x
s
b r
s

0 1
y-intercept: b y b x
 

Section 10.3-10
Example
Let us return to the example from Section 10.2. We
would like to use the explanatory variable, x, shoe print
length, to predict the response variable, y, height.
The data are listed below:

Section 10.3-11
Example - Continued
Requirement Check:
1. The data are assumed to be a simple random
sample.
2. The scatterplot showed a roughly straight-line
pattern.
3. There are no outliers.
The use of technology is recommended for finding the
equation of a regression line.

Section 10.3-12
Example – Continued

Section 10.3-13
Example - Continued
All these technologies show that the regression equation
can be expressed as:
Now we use the formulas to determine the regression
equation (technology is recommended).
ˆ 125 1.73
y x
 

Section 10.3-14
Example
Recall from the previous section that r = 0.591269.
Technology can be used to find the values of the sample
means and sample standard deviations used below.
(These are the same coefficients found using
technology)
  
1
0 1
4.87391
0.591269 1.72745
1.66823
177.3 1.72745 30.04 125.40740
y
x
s
b r
s
b y b x
  
    

Section 10.3-15
Example
Graph the regression equation on a scatterplot:
ˆ 125 1.73
y x
 

Section 10.3-16
1. Use the regression equation for predictions only if the
graph of the regression line on the scatterplot confirms
that the regression line fits the points reasonably well.
2. Use the regression equation for predictions only if the
linear correlation coefficient r indicates that there is a
linear correlation between the two variables (as
described in Section 10-2).
Using the Regression
Equation for Predictions

Section 10.3-17
3. Use the regression line for predictions only if the data do
not go much beyond the scope of the available sample
data. (Predicting too far beyond the scope of the
available sample data is called extrapolation, and it
could result in bad predictions.)
4. If the regression equation does not appear to be useful
for making predictions, the best predicted value of a
variable is its sample mean.

Section 10.3-18
Strategy for Predicting Values of y

Section 10.3-19
If the regression equation is not a good model, the best
predicted value of y is simply , the mean of the y
values.
Remember, this strategy applies to linear patterns of
points in a scatterplot.
If the scatterplot shows a pattern that is not a straight-line
pattern, other methods apply, as described in Section 10-
6.
y

Section 10.3-20
Example
Use the 5 pairs of shoe print lengths and heights to
predict the height of a person with a shoe print length of
29 cm.
The regression line does not fit the points well. The
correlation is r = 0.591, which suggests there is not a
linear correlation (the P-value was 0.294).
The best predicted height is simply the mean of the
sample heights:
177.3 cm
y 

Section 10.3-21
Example
Use the 40 pairs of shoe print lengths from Data Set 2 in
Appendix B to predict the height of a person with a shoe
print length of 29 cm.
Now, the regression line does fit the points well, and the
correlation of r = 0.813 suggests that there is a linear
correlation (the P-value is 0.000).

Section 10.3-22
Example - Continued
Using technology we obtain the regression equation and
scatterplot:

Section 10.3-23
Example - Continued
The given shoe length of 29 cm is not beyond the scope
of the available data, so substitute in 29 cm into the
regression model:
A person with a shoe length of 29 cm is predicted to be
174.3 cm tall.
 
ˆ 80.9 3.22
80.9 3.22 29
174.3 cm
y x
 
 


Section 10.3-24
Part 2: Beyond the Basics of Regression

Section 10.3-25
Definition
In working with two variables related by a regression
equation, the marginal change in a variable is the amount
that it changes when the other variable changes by
exactly one unit.
The slope b1 in the regression equation represents the
marginal change in y that occurs when x changes by one
unit.

Section 10.3-26
Example
For the 40 pairs of shoe print lengths and heights, the
regression equation was:
The slope of 3.22 tells us that if we increase shoe print
length by 1 cm, the predicted height of a person
increases by 3.22 cm.
ˆ 80.9 3.22
y x
 

Section 10.3-27
Definition
In a scatterplot, an outlier is a point lying far away from
the other data points.
Paired sample data may include one or more influential
points, which are points that strongly affect the graph of
the regression line.

Section 10.3-28
Example
For the 40 pairs of shoe prints and heights, observe what
happens if we include this additional data point:
x = 35 cm and y = 25 cm

Section 10.3-29
Example - Continued
The additional point is an influential point because the
graph of the regression line because the graph of the
regression line did change considerably.
The additional point is also an outlier because it is far
from the other points.

Section 10.3-30
For a pair of sample x and y values, the residual is the
difference between the observed sample value of y and
the y-value that is predicted by using the regression
equation.
That is:
Definition
ˆ
residual observed predicted
y y y y
   

Section 10.3-31
Residuals

Section 10.3-32
A straight line satisfies the least-squares property if the
sum of the squares of the residuals is the smallest sum
possible.
Definition

Section 10.3-33
A residual plot is a scatterplot of the (x, y) values after
each of the y-coordinate values has been replaced by
the residual value y – ŷ (where ŷ denotes the predicted
value of y).
That is, a residual plot is a graph of the points (x, y – ŷ).
Definition

Section 10.3-34
Residual Plot Analysis
When analyzing a residual plot, look for a pattern in the
way the points are configured, and use these criteria:
The residual plot should not have any obvious patterns
(not even a straight line pattern). This confirms that the
scatterplot of the sample data is a straight-line pattern.
The residual plot should not become thicker (or thinner)
when viewed from left to right. This confirms the
requirement that for different fixed values of x, the
distributions of the corresponding y values all have the
same standard deviation.

Section 10.3-35
Example
The shoe print and height data are used to generate the
following residual plot:

Section 10.3-36
Example - Continued
The residual plot becomes thicker, which suggests that
the requirement of equal standard deviations is violated.

Section 10.3-37
Example - Continued
On the following slides are three residual plots.
Observe what is good or bad about the individual
regression models.

Section 10.3-38
Example - Continued
Regression model is a good model:

Section 10.3-39
Example - Continued
Distinct pattern: sample data may not follow a straight-
line pattern.

Section 10.3-40
Example - Continued
Residual plot becoming thicker: equal standard
deviations violated.

Section 10.3-41
Complete Regression Analysis
1. Construct a scatterplot and verify that the pattern of
the points is approximately a straight-line pattern
without outliers. (If there are outliers, consider their
effects by comparing results that include the outliers
to results that exclude the outliers.)
2. Construct a residual plot and verify that there is no
pattern (other than a straight-line pattern) and also
verify that the residual plot does not become thicker
(or thinner).

Section 10.3-42
Complete Regression Analysis
3. Use a histogram and/or normal quantile plot to
confirm that the values of the residuals have a
distribution that is approximately normal.
4. Consider any effects of a pattern over time.

Chapter 10 Section 3.ppt

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