3.2 least squares regression part i: interpreting a
TRANSCRIPT
3.2
Least Squares Regression
Part I: Interpreting a Regression Line & Prediction
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.
Plot the data
Look for overall pattern (DOFS)
Calculate numerical summaries (r)
When data falls into a regular pattern seek for a simplified model
Mathematical relationship between two quantitative
variables
How does a response variable y change with respect
to x changes?
Can you make predictions of y for a given x?
Is it a good regression model?
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)
0 s 4 s 8 s 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
Suppose: ◦ y -> response variable (on vertical axis)
◦ x -> explanatory variable (on the horizontal axis).
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:
PROBLEM: Identify the slope and y
intercept of the regression line.
Interpret each value in context.
price = 38257-0.1629(miles driven)
Miles driven
Pri
ce
(in
do
llars
)
160000140000120000100000800006000040000200000
45000
40000
35000
30000
25000
20000
15000
10000
5000
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 F-
150 with 100,000 miles driven.
price = 38257-0.1629(miles driven)
What are Regression Lines used for?
Accuracy?
Can you predict a response ŷ for any of the
explanatory variable x?
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. Check your Understanding Page 168
2. 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
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