Sections 2'3 and 2'4

1
Sections 2.3 and 2.4

• Regression Lines

2
Review of Lines

• The equation yabx describes a line.
• a is the y-intercept or the value of y that goes
  with x0.
• b is the slope of the line.
• ab is the value of y that goes with x1.
• a2b is the value of y that goes with x 2.
• When the value of x goes up by one, the value of
  y goes up by b.
• Plot a line, by first plotting two points.

3
Used Honda Civics
4
Can mileage be used to predict price?

• Sketch line.
• Estimate slope
• Estimate y-intercept.
• Give equation of line.
• Estimate price of car with 100,000 miles.

5
Regression Lines

• Can we agree on the best line?
• The least squares regression line is the line
  that best fits.
• Goodness of fit is determined by calculating the
  sum of all squared vertical distances. The
  smaller this sum, the better the fit.
• The regression line is given by yabx, where

6
Back to Civics

• Use the formulas for finding least squares
  regression line. The descriptive statistics for
  our data are as follows
• Describe what the slope and y-intercept mean in
  the vocabulary of the problem.
• Does the y-intercept make sense in the context of
  the data?

7
Extrapolation

• extrapolation using the regression line for
  prediction far outside the range of values of the
  explanatory variable used to obtain the line.
  Such predictions are often not accurate.
• Usually in regression x0 is outside range of
  data so the y-intercept will not make sense.
• Extrapolation would also occur if we tried to
  predict the price of a used Civic with 300,000
  miles.

8
How successful is the regression in explaining
the response?

• r2 - the proportion of the variation in the
  response variable that is explained by the
  regression line.

Observed Prices Leaf Unit 100 1 7 3
1 8 2 9 9 2 10 (2) 11 99
2 12 9 1 13 9 Mean 11379 S2347
Predicted Prices Leaf Unit 100 1 7
2 1 8 1 9 2 10 1 2 11
(2) 12 25 2 13 00 Mean 11379 S2295
9
Residuals

• The residual is the vertical distance from the
  line (positive if above line, negative if below
  line.)
• Find the residual for the car with 43903 miles.

10
How well does line fit data?

• Look at a residual plot.
• residual observed y predicted
• plot explanatory variable on x-axis residuals on
  y- axis.
• If the regression captured the overall pattern,
  then there should be no pattern remaining in
  residual plot.
• How well does the least squares regression line
  fit the relationship between mileage and price?

11
Outliers

• An outlier is a data point that falls outside the
  overall pattern. They are most obvious from
  looking at the residual plot.
•  
• Are there any outliers in the previous example?

12
Influential Observations

• An observation is influential if removing it
  markedly changes the calculated regression line.
• Example on Web
• Points that are separated from the data in the x
  direction of a scatterplot tend to be
  influential. (These points are may not be
  outliers as defined on previous slide.)
• In the previous example, which point may be
  influential?

13
Influential Observations, Continued

• The equations with and without this point
  are PRICE 15589.6 - 0.0697811 MILEAGE PRICE
  15990.5 - 0.0786652 MILEAGE
• How far apart are these?
• We would predict the price of a car with 30,000
  miles to be 13496 vs 13631
• We would predict the price of a car with 70,000
  miles to be 10705 vs 10484
• Sketch Line

14
Summary

• Get equation of regression line.
• If data is given, use calculator or Minitab.
• If summary statistics are given use formula.
• Interpret slope and y-intercept in context of
  problem.
• r2 gives proportion of variation in response
  variable that is explained by the explanatory
  variable via the line
• Residual plots are used to check for outliers and
  pattern that is not picked up by line.
• Outliers vs Influential data points.

15
Year as an Explanatory Variable
16
Residual Plot
17
Fitting Quadratic
18
Example. SAT Math vs SAT Verbal

• States average SAT math scores against states
  average SAT verbal scores.
• verbal 34.2 0.940 math
• r 0.970 (r2 0.941).

19
Example. SAT Math vs SAT Verbal

• Describe what the slope and y-intercept mean in
  the vocabulary of the problem.
• How well does the line fit the data?
• How successful is the regression in explaining
  response?
• How well would this line do to predict individual
  verbal scores ?

20
Words of Caution

• Extrapolation
• Using Averaged Data
• Lurking Variables
• Association Does Not Imply Causation
• Example. Among elementary aged children there is
  a strong correlation between shoe size and
  reading aptitude. What explains this correlation?