Stat 401 B – 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
2
Linear Fit
Linear Fit
yˆ = βˆ0 + βˆ1 x
„Predicted Temp = 9.8815 +
0.012584*CO2
3
1
Stat 401 B – 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 401 B – Lecture 7
Step 1 - Hypotheses
H 0 : β1 = 0
H A : β1 ≠ 0
7
Step 2 – Test Statistic
t=
(βˆ − 0)
1
se( βˆ1 )
se( βˆ1 ) =
MS Error
2
∑ (x − x )
df = n − 2
8
Step 2 – Test Statistic
βˆ1 = 0.012584
se( βˆ1 ) = 0.001456
βˆ − 0 0.012584
=
= 8.64
t= 1
se( βˆ ) 0.001456
(
)
1
P - value < 0.0001
9
3
Stat 401 B – Lecture 7
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.
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Confidence Interval
( )
βˆ1 ± t * se βˆ1
t * from t - table with df = n − 2
95% confidence, df = 18,
t * = 2.101
12
4
Stat 401 B – Lecture 7
Confidence Interval
( )
βˆ1 ± t * se βˆ1
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.
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
5
Stat 401 B – 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, β̂ 0 .
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Test of Hypotheses
H 0 : β0 = 0
: β0 ≠ 0
βˆ 0 − 0
t =
se ( βˆ )
H
A
(
)
0
df = n − 2
17
Confidence Interval
βˆ0 ± t * se( βˆ0 )
t * from t - table
with df = n − 2
18
6