Linear Regression
The Science of
Predicting Outcome
Least-Squares
Regression
LSR is a method for finding a line that
summarizes the relationship between
two variables
Regression line is a straight line that
describes how a response variable y
changes as an explanatory variable x
changes
We often use a regression line to predict
the value of y for a given value of x
LSRL: Least Square
Regression Line
Y-intercept
Slope
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.)
7
10
12
8
9.5
10.5
11
12.5
13.5
10
Height (in)
64
69
71
68
71
70
72
74
77
68
Example #1 - Finding the LSRL
We need our
graphing
calculator to
solve the first
Case for today
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
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
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
(this is a prediction and is therefore not exact.)
Practice
Student
1
2
A. Find the strength of correlation
between the 2 variables
3
6
7
B. Write the linear model for this
data set
9
11
13
4
C. What will be your BAC level if
you drink 6 bottle of beers.
5
8
10
12
14
15
16
Number
of Beers
5
2
9
7
3
3
4
5
8
3
5
5
6
7
1
4
Blood Alcohol
Level
0.1
0.03
0.19
0.095
0.07
0.02
0.07
0.085
0.12
0.04
0.06
0.05
0.1
0.09
0.01
0.05
Coefficients a and b
The slope is:
The intercept is:
S-sub y and s-sub x are the
sample standard deviations of
y and x (kinda like rise over
run)
y-bar and x-bar are the
mean y and x respectively
The equation of the least squares regression line
is written as:
This table describes a study that recorded data on number of beers
consumed and blood alcohol content (BAC) for 16 students. Here is
some partial computer output from Minitab relating to these data:
Y-intercept
Slope
(a) Use the computer output to write the equation of the least-squares
line.
(b) Interpret the slope and y intercept of the equation in this setting.
(c) What blood alcohol level would your equation predict for a student
who consumed 6 beers?
Answers
(a) If y = blood alcohol content (BAC) and x = number of beers,
BAC = −0.01270 + 0.017964(number of beers).
(b) Slope: for every extra beer consumed, the BAC will increase by an
average of 0.017964.
Intercept: if no beers are consumed, the BAC will be, on average,
−0.01270 (obviously meaningless).
(c) Predicted BAC = 0.0951
Here’s a computer generated output of 2
bivariate data. Write a linear model that
corresponds to these set of data.
y-hat = -0.124 + 0.0179(x)
Class Activity:
Arm-span vs Height
“On predicting height given arm span
“
Students will measure their height and arm span.
Then they will write the LSRL from the data they
collected and predict a person’s arm span with their
height.