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## Chapter 8: Linear Regression

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### Chapter 8: Linear Regression A.P. Statistics Least-Squares Regression Line We Can Find the LSRL For Three Different Situations Using z-Scores of Real Data ... – PowerPoint PPT presentation

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Title: Chapter 8: Linear Regression

1
Chapter 8 Linear Regression
• A.P. Statistics

2
Linear Model
• Making a scatterplot allows you to describe the
relationship between the two quantitative
variables.
• However, sometimes it is much more useful to use
that linear relationship to predict or estimate
information based on that real data relationship.
• We use the Linear Model to make those predictions
and estimations.

3
Linear Model
• Normal Model
• Linear Model
• Allows us to make predictions and estimations
about the population and future events.
• It is a model of real data, as long as that data
has a nearly symmetric distribution.
• Allow us to make predictions and estimations
about the population and future events.
• It is a model of real data, as long as that data
has a linear relationship between two
quantitative variables.

4
Linear Model and the Least Squared Regression Line
• To make this model, we need to find a line of
best fit.
• This line of best fit is the predictor line and
will be the way we predict or estimate our
response variable, given our explanatory
variable.
• This line has to do with how well it minimizes
the residuals.

5
Residuals and the Least Squares Regression Line
• The residual is the difference between the
observed value and the predicted value.
• It tells us how far off the models prediction is
at that point
• Negative residual predicted value is too big
(overestimation)
• Positive residual predicted value is too small
(underestimation)

6
Residuals
7
Least Squares Regression Line
• The LSRL attempts to find a line where the sum of
the squared residuals are the smallest.
• Why not just find a line where the sum of the
residuals is the smallest?
• Sum of residuals will always be zero
• By squaring residuals, we get all positive
• Emphasizes the large residualswhich have a big
impact on the correlation and the regression line

8
Scatterplot of Math and Verbal SAT scores
9
Scatterplot of Math and Verbal SAT scores with
incorrect LSRL
10
Scatterplot of Math and Verbal SAT scores with
correct LSRL
11
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12
Least-Squares Regression Line
• We Can Find the LSRL For Three Different
Situations
• Using z-Scores of Real Data (Standardizing Data)
• Using Summary Statistics of Data (mean and
standard deviation)
• Using Real Data

13
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14
LSRL Using z-Scores of Real Data
• LSRL passes through
• and
• LSRL equation is
• moving one standard deviation from the mean
in x, we can expect to move about r standard
deviations from the mean in y .

15
LSRL Using z-Scores of Real Data
(Interpretation)
• LSRL of scatterplot
• For every standard deviation above (below) the
mean a sandwich is in protein, well predict that
that its fat content is 0.83 standard deviations
above (below) the mean.

16
LSRL Using Summary Statistics of Data
Protein Fat
LSRL Equation
17
LSRL Using Summary Statistics of Data
(Interpretation)
Slope One additional gram of protein is
associated with an additional 0.97 grams of
fat. y-intercept An item that has zero grams
of protein will have 6.8 grams of fat.
ALWAYS CHECK TO SEE IF Y-INTERCEPT MAKES SENSE IN
THE CONTEXT OF THE PROBLEM AND DATA
18
LSRL Using Summary Statistics of Data
(Interpretation)
• Use technology to get the LSRL. Making sure you

19
Properties of the LSRL
• The fact that the Sum of Squared Errors (SSE,
same as Least Squared Sum)is as small as possible
means that for this line
• The sum and mean of the residuals is 0
• The variation in the residuals is as small as
possible
• The line contains the point of averages

20
Assumptions and Conditions for using LSRL
• Quantitative Variable Condition
• Straight Enough Condition
• if notre-express (Chapter 10)
• Outlier Condition
• with and without ?

21
Residuals and LSRL
• Residuals should be used to see if a linear model
is appropriate
• Residuals are the part of the data that has not
been modeled in our linear model

22
Residuals and LSRL
• What to Look for in a Residual Plot to Satisfy
Straight Enough Condition
• No patterns, no interesting features (like
direction or shape), should stretch horizontally
with about same scatter throughout, no bends or
outliers.
• The distribution of residuals should be symmetric
if the original data is straight enough.

Looking at a scatterplot of the residuals vs. the
x-value is a good way to check the Straight
Enough Condition, which determines if a linear
model is appropriate.
23
Residuals, again
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27
A Complete Linear Regression AnalysisPART I
• Draw a scatterplot of the data. Comment on what
you see. (Satisfy Quantitative Data Condition)
• Form, strength, direction
• Unusual Points, Deviations
• Comment on General Variable Direction

28
A Complete Linear Regression AnalysisPART II
• Compute r . Comment on what r means in context
and if it is appropriate to use (does the
relationship seem linearStraight Enough
Condition)

29
A Complete Linear Regression AnalysisPART III
• Find the LSRL
• Check all three conditions
• Quantitative Data Condition
• Straight Enough Condition
• Outlier Condition

30
A Complete Linear Regression AnalysisPART IV
• Draw a residual plot and interpret it-is the
linear model appropriate?

31
A Complete Linear Regression AnalysisPART V
• Interpret slope in context
• Interpret the y-intercept in context

32
A Complete Linear Regression AnalysisPART VI
• Compute R-Squared. Interpret the value and use
as a measure for the accuracy of the model. How
well does the model predict?

33
What is R-Squared
• This value will determine how accurate the linear
model is predicting your y-values from you
x-values.
• It is written as a percent.
• It is, literally, your r-value squared.

34
R-Squared Interpretation
• If a Regression analysis has an R-squared value
of 97, that means the model does an excellent
job predicting the y-values in your model.
• How do we interpret that?
• 97 of the variation is y can be accounted for
by the variation is x, on average.

35
R-Squared Interpretation
• There are other ways to write that
interpretation.
• Also, can be thought of as
• how much error was eliminated in our predictions
if we used the LSRL instead of a guess of .