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