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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
    values, which can be added
  • 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
    check your conditions, etc.

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 .
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