Transcript PowerPoint

2 Variable Stats
If you eat more
junk food will you
gain weight?
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By using a scatter plot we can view a
visual pattern in the graph
This pattern can reveal the nature of
the relationship between the two
variables
Weight Change after Exercise
Weight (kg)
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Time ( months )
Variables
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When Graphing this information you
must determine which variable goes
on which axis
One Variable is the dependent (or
response) variable
The other is the independent (or
explanatory) variable
Which Variable goes Where?
y
Independent
x
Independent or Dependent
Variable?
For each relationship, sketch the axis
and label the independent and
dependent variables
a)
b)
c)
The number of hours a student studies per course
and his or her average mark in the course
The time of day and the number of cups of coffee
sold
The amount of snowfall and the number of collisions
on major highways
Correlation
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Variables have a LINEAR
CORRELATION if changes in
one variable tend to be
proportional to the changes
in another variable
If your Y increases at a
constant rate as X increases
then you have a Perfect
Positive (or direct) linear
correlation
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If the Y decreases at a constant rate
as X increases then you have a
Perfect Negative (or inverse)
linear correlation
Strength of Correlation
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The stronger the correlation the
more closely the data points cluster
around the line of best fit.
• State the strength *strong, weak,
moderate, moderately strong etc….*
• State direction *positive or negative*
• State type of correlation *Linear, Nonlinear, exponential etc…*
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Turn to page 160 in your textbook
Example 1 Classifying Linear
Relationships
Correlation Coefficient
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Karl Pearson
(1857-1936)
developed a
formula to
determine a value
that would tell us
the strength of a
correlation.
Note he also
developed the
standard deviation
Correlation Covariance
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First need to define what we call the
covariance of two variables in a
sample.
s XY
1

( x  x)( y  y )
n 1
Correlation Coefficient
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Denoted with an r
Is the covariance divided by the
product of the standard deviations
for X and Y
s XY
r
sX  sY
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Gives a quantitative measure of the
strength of a relationship
Basically shows how close the data
points cluster around the line of best
fit.
Also called the Pearson productmoment coefficient of correlation
(PPMC) or Pearson’s r
Strength of Correlation
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The closer the correlation coefficient
(r ) is to 1 or -1 the stronger the
correlation
r = -1
r =+1
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No relation has a correlation that is
close to zero and the dots are
scattered everywhere
r=0
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The following diagram illustrates how
the correlation coefficient
corresponds to the strength of a
linear correlation
Negative Linear Correlation
Perfect
Strong
-1
Moderate
-0.67
Positive Linear Correlation
Weak
Weak
-0.33
0
Moderate
0.33
0.67
Perfect
Strong
1
SO LONG!!!!!!!!
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There is a quicker way to calculate r,
without having to find the standard
deviations……but it looks more
difficult
r
n  xy  ( x)( y)
[n  x  ( x) ][n  y  ( y) ]
2
2
2
2
Example
Mean Temp (°C)
A Farmer wants to
determine whether
there is a relationship 4
between the mean
8
temperature during
the growing season 10
and the size of his
9
wheat crop. He
assembles the
following data for the 11
last six crops
6
Yield
(tonnes/hectare)
1.6
2.4
2.0
2.6
2.1
2.2
a) Does the Scatter plot of this
data indicate any linear
correlation between the two
variables
2
1.5
1
0.5
0
Yield (T/ha)
2.5
Yield of Growing Season
0
2
4
6
8
10
12
14
Mean Temperature (ºC)
b) Compute the Correlation Coefficient
Temp
Yield
4
1.6
8
2.4
10
2.0
9
2.6
11
2.1
6
2.2
x
y
X2
 x2
Y2
 y2
xy
 xy
r
n  xy  ( x)( y)
[n  x2  ( x)2 ][n  y 2  ( y)2 ]
c) What can the farmer conclude about the
relationship between the mean temperatures
during the growing season and the wheat yields on
his farm
Homework
Pg 168 #1-6