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
2
R
(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
Step 1 - Hypotheses
H 0 : 1  0
H A : 1  0
7
Step 2 – Test Statistic
Test
Statistic:
Std Error:

ˆ  0 
t
1
se( ˆ1 )
MS
Error
ˆ
se( 1 ) 
2
 x  x 
df  n  2
8
Step 2 – Test Statistic
ˆ1  0.012584
se( ˆ1 )  0.001456
ˆ1  0 0.012584
t

 8.64
se( ˆ ) 0.001456


1
P - value  0.0001
9
Step 3 - Decision
Reject the null hypothesis
because the P-value is so
small (smaller than 0.05).
10
Step 4 – Conclusion
Based on our sample data,
there is a statistically
significant linear relationship
between annual carbon
dioxide concentration and
annual global temperature.
11
Confidence Interval
 
*
ˆ
1  t se ˆ1
t from t - table with df  n  2
95% confidence , df  18,
*
t  2.101
*
12
Confidence Interval
 
*
ˆ
ˆ
1  t se 1
0.012584  2.1010.001456 
0.012584  0.003059
0.0095 to 0.0156
13
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.
14
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.
15
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 .
16
Test of Hypotheses
H 0 : 0  0
H A : 0  0
ˆ0  0
t
ˆ
se(  )


0
df  n  2
17
Confidence Interval
*
ˆ
ˆ
 0  t se(  0 )
*
t from t - table
with df  n  2
18