3.2
Least Squares Regression
Part I: Interpreting a Regression Line & Prediction
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INTERPRET the slope and y intercept of a least-squares
regression line.
USE the least-squares regression line to predict y for a given x.
CALCULATE and INTERPRET residuals and their standard
deviation.
EXPLAIN the concept of least squares.
DETERMINE the equation of a least-squares regression line
using a variety of methods.
CONSTRUCT and INTERPRET residual plots to assess whether
a linear model is appropriate.
ASSESS how well the least-squares regression line models the
relationship between two variables.
DESCRIBE how the slope, y intercept, standard deviation of the
residuals, and r 2 are influenced by outliers.
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Plot the data
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Look for overall pattern (DOFS)
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Calculate numerical summaries (r)
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When data falls into a regular pattern seek for
a simplified model
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Mathematical relationship between two quantitative
variables
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How does a response variable y change with respect
to x changes?
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Can you make predictions of y for a given x?
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Is it a good regression model?
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Describe the graph:
Don’t you hate it when you open a can of soda and some of the contents
spray out of the can?
Two AP®Statistics students, Kerry and Danielle, wanted to investigate if
tapping on a can of soda would reduce the amount of soda expelled after the
can has been shaken.
For their experiment, they vigorously shook 40 cans of soda and randomly
assigned each can to be tapped for 0 seconds, 4 seconds, 8 seconds, or 12
seconds.
Then, after opening the can and cleaning up the mess, the students measured
the amount of soda left in each can (in ml). Here are the data and a
scatterplot.
The scatterplot shows a fairly strong, positive linear association between the
amount of tapping time and the amount remaining in the can. The line on the
plot is a regression line for predicting the amount remaining from the amount
of tapping time.
Amount of soda remaining (ml)
0s
4s
8s
12 s
245
260
267
275
255
250
271
280
250
250
268
275
250
250
270
280
250
260
276
285
245
265
255
290
248
267
270
284
250
260
270
278
251
261
275
279
249
259
275
280
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Suppose:
◦ y -> response variable (on vertical axis)
◦ x -> explanatory variable (on the horizontal axis).
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A regression line equation:
 ŷ = a + bx
• ŷ (read “y hat”) ≡ predicted value of the response variable
y for a given value of the explanatory variable x.
• b ≡ slope, the amount by which y is predicted to change
when x increases by one unit.
• a ≡ y intercept, the predicted value of y when x = 0.
Regression Line equation:
price = 38257- 0.1629(miles driven)
45000
PROBLEM: Identify the slope and y
intercept of the regression line.
40000
Price (in dollars)
35000
30000
25000
Interpret each value in context.
20000
15000
10000
5000
0
20000
40000
60000
80000 100000
Miles driven
120000
140000
160000
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The equation of the regression line in the
previous Alternate Example is
soda = 248.6 +2.63 (tapping time)
Problem: Identify the slope and y intercept of
the regression line. Interpret each value in
context.
Use the regression line to predict price for a Ford F150 with 100,000 miles driven.
price = 38257- 0.1629(miles driven)
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What are Regression Lines used for?
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Accuracy?
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Can you predict a response ŷ for any of the
explanatory variable x?
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What is considered a Good Regression Line?
A good regression line makes the vertical distances
of the points from the line as small as possible.
residual
residual = observed y – predicted y
residual = y – ŷ
If the
residual is
+ve/-ve?
Common Errors:
1. Not stating that the slope is the predicted
(estimated/expected value) change in the y
variable for each increase/decrease of 1 unit in
the x variable.
Is it accurate to say the following? Explain:
“The price will go down by 0.1629 dollars for each
additional mile driven”
1.
2.
Check your Understanding Page 168
Activity: page 170 Investigating Properties
of the Least Square Regression Line
LiST THEM on the whiteboard
3.
Page 171: Least Squares Regression Lines
on the Calculator
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Brainstorm with your group and provide a
quick summary on the white board
Have examples to ensure your understanding
on the topic
Homework: page 193 # 35-42 ALL