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60
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Scatterplot of Weight vs. Height
r = 0.74
Height
Weight
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8 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
Summary
There is a linear association between
x
and
y
(height and
weight)
The correlation coefficient tells us about the strength of the
relationship
We want to know more–describing
y
(weights), based on
x
(heights)
This technique–using information about one variable to
predict the value of another, based on a linear relationship–is
known as regression
Best way to do this–line through averages of groups
But which groups? How do we generalize this?
9 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
The Formula
Problem: We want to find the line that smooths the averages of
groups.
Solution: Match up the SDs, and account for the correlation.
Associated with a one SD increase in
x
is an
r
SD increase in
y
, on average:
y

¯
y
SD
y
=
r
x

¯
x
SD
x
This is the regression equation
10 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
Example: Averages
Data is collected on the heights and weights of adults. It is found
that the average height is 65 inches, with an SD of 2 inches, while
the average weight is 140 pounds, with an SD of 10 pounds.
Height and weight have a correlation coefficient of 0.74.
Find the average weight of people with with height 62 inches.
1
Decide which variable is
x
(explanatory/independent) and
which variable is
y
(response/dependent):
Since we are given height, it is the
x
, while weight, which we
want to predict, is the
y
variable
11 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
Example: Averages (cont.)
2
Calculate
y
in standard units:
Since we are given the
x
means and SDs, plug them into the
regression equation.
y

¯
y
SD
y
=
r
x

¯
x
SD
x
y

140
10
= 0
.
74
62

65
2
=

1
.
11
12 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
Example: Averages (cont.)
3
Solve for
y
, in its desired units:
Since we desire
y
in its original units, we convert the standard
units into pounds
y

140
10
=

1
.
11
y

140 =

11
.
10
y
=

11
.
10 + 140
= 128
.
9
4
Interpret the answer:
People with height 62 inches weigh, on average, 128.9 pounds
13 / 34
Today’s Goals
The Intuition
The Mechanics
Some Caveats
Simplification
Example: Individuals
Data is collected on the heights and weights of adults. It is found
that the average height is 65 inches, with an SD of 2 inches, while
the average weight is 140 pounds, with an SD of 10 pounds.