Stat 301 – Lecture 7
Simple Linear Regression
Question
Is
annual carbon dioxide
concentration related to
annual global temperature?
1
Bivariate Fit of Temp By CO2
15.0
Temp
14.5
14.0
13.5
300
350
400
CO2
Linear Fit
2
Linear Fit
yˆ  ˆ0  ˆ1 x
Predicted Temperature =
9.8815 + 0.012584*CO2
3
1
Stat 301 – Lecture 7
R2 (RSquare)
80.6% of the variation in the
global temperature can be
explained by the linear
relationship with carbon
dioxide concentration.
19.4% is unexplained.
4
Interpretation
There is a fairly strong
positive linear relationship
between annual carbon
dioxide concentration and
annual global temperature.
5
Testing Hypotheses
Question
Is
the linear relationship
between annual carbon
dioxide concentration and
annual global temperature
statistically significant?
6
2
Stat 301 – Lecture 7
Step 1 - Hypotheses
H 0 : 1  0
H A : 1  0
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Step 2 – Test Statistic
ˆ  0
Test
Statistic:
t
Std Error:
se( ˆ1 ) 
This image cannot currently be display ed.
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se( ˆ1 )
MS Error
2
 x  x 
df  n  2
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Step 2 – Test Statistic
ˆ1  0.012584
se( ˆ1 )  0.001456
ˆ  0 0.012584

 8.64
t 1
se( ˆ ) 0.001456
This image cannot currently be display ed.


1
P - value  0.0001
9
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Stat 301 – Lecture 7
Step 3 - Decision
Reject the null hypothesis
because the P-value is so
small (smaller than 0.05).
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Step 4 – Conclusion
Based on our sample data,
there is a statistically
significant linear relationship
between annual carbon
dioxide concentration and
annual global temperature.
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Confidence Interval
 
ˆ1  t * se ˆ1
t * from t - table with df  n  2
95% confidence, df  18,
t *  2.101
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Stat 301 – Lecture 7
Confidence Interval
 
ˆ1  t * se ˆ1
0.012584  2.1010.001456 
0.012584  0.003059
0.0095 to 0.0156
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Interpretation – Part 1
The population slope
parameter relating CO2 to
temperature could be any
value between 0.0095 and
0.0156.
Units are o C/ppmv.
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Interpretation – Part 2
We are 95% confident that
intervals based on random
samples from the population
with capture the actual
population slope parameter.
This is confidence in the
process.
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5
Stat 301 – Lecture 7
Y-Intercept
When the Y-intercept is
interpretable within the
context of the problem, it may
be of interest to do inference
based on the estimated yintercept, ̂ 0 .
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Test of Hypotheses
H 0 : 0  0
: 0  0
ˆ 0  0
t 
se ( ˆ )
H
A


0
df  n  2
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Confidence Interval
ˆ0  t * se( ˆ0 )
t * from t - table
with df  n  2
18
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