Least Squares Regression Line (LSRL)

Introduction

• Many times the scatterplot shows some pattern in the data.

• For now, we will look at the analysis of data that falls in a straight line pattern.

Size of Diamond vs. Price

Introduction

• When we see a straight line pattern, we want to model the data with a linear equation.

• This will allow us to make predictions and actually use our data.

Linear Relations•We know lines from algebra to come in the form y = mx + b, where m is the slope and b is the y-intercept.

•In statistics, we use y = a + bx for the equation of a straight line. Now a is the intercept and b is the slope.

•The slope (b) of the line, is the amount by which y increases when x increase by 1 unit.

– This interpretation is very important.

•The intercept (a), sometimes called the vertical intercept, is the height of the line when x = 0.

Example

• Consider the equation: y=7+3x– The slope is 3.

• For every increase of 1 in the x-variable, there will be an increase of 3 in the y-variable.

– The intercept is 7.• When the x-variable is

0, the y-variable is 7.x

y

0 2 4 6 8

0

5

10

15 y = 7 + 3x

a = 7

x increases by 1

y increases by b = 3

Example

• Consider the equation: y=17-4x– The slope is -4.

• For every increase of 1 in the x-variable, there will be a decrease of 4 in the y-variable.

– The intercept is 17.• When the x-variable is

0, the y-variable is 17.

y

y = 17 - 4x

x increases by 1

y changes by b = -4 (i.e., changes by –4)

0 2 4 6 8

0

5

10

15

a = 17

Least Squares Line

• How can we find the best line to fit the data?– We would like to minimize the total distance

away from the line• This distance is measured vertically from the point

to the line.

– Go to the following applet and start plotting points to see how this process works.

Make your own regression line

Least Squares Line

• You first get a line once you plot two points.• When you plot the third, green bars appear

representing the error (actually called residual) of the line.– These are how far off your line is for each of the

points.

• The best line is the one that would minimize the total length of the green lines (all put together).

Guess the best fit line

• Go to the following applet to practice your skills at estimating an LSRL.– Plot a bunch of points.– Then click the draw line button and draw what

you think is the best fit line– Then, check the “show least squares line”

checkbox

To the applet

The details of the LSRL

• The mathematics involved in calculating the LSRL is a bit complicated.

Coefficients a and b

bxay ˆThe equation of the least squares regression line is written as:

The little symbol above the y is a hat! The equation is read as, “y-hat equals a plus bx.” The ‘y-hat’ indicates that this is a regression line and that the model (equation) is to be used to make predictions.

The slope is:

x

y

s

srb

The intercept is: xbya y-bar and x-bar are the mean y and x respectively

S-sub y and s-sub x are the sample standard deviations of y and x (kinda like rise over run)

Three Important Questions • To examine how useful or effective the line

summarizing the relationship between x and y, we consider the following three questions.1. Is a line an appropriate way to summarize the

relationship between the two variables?

2. Are there any unusual aspects of the data set that we need to consider before proceeding to use the regression line to make predictions?

3. If we decide that it is reasonable to use the regression line as a basis for prediction, how accurate can we expect predictions based on the regression line to be?

Example #1 - Finding the LSRL

• Consider the following data:

• With this data, find the LSRL

• Start by entering this data into list 1 and list 2

Shoe Size (men’s U.S.) Height (in)

7 64

10 69

12 71

8 68

9.5 71

10.5 70

11 72

12.5 74

13.5 77

10 68

Example #1 - Finding the LSRL

• You should then see the results of the regression.– a=53.24– b=1.65– r-squared=.8422– r=.9177

)(65.124.53

65.124.53ˆ

sizeshoeHeight

xy

This is the correlation coefficient for the scatterplot!!!

Example #2 – Interpreting LSRL

• Interpreting the intercept– When your shoe size is 0, you should be about 53.24

inches tall– Of course this does not make much sense in the

context of the problem• Interpreting the slope

– For each increase of 1 in the shoe size, we would expect the height to increase by 1.65 inches

)(65.124.53

65.124.53ˆ

sizeshoeHeight

xy

Example #3 – Using LSRL

• Making predictions– How tall might you expect someone to be who has a

shoe size of 12.5?– Just plug in 12.5 for the shoe size above, so…– Height = 53.24+1.65 (12.5)=73.865 inches

• Of course this is a prediction and is therefore not exact.

)(65.124.53

65.124.53ˆ

sizeshoeHeight

xy

Least Squares Regression Line (LSRL)

• This concludes this presentation.