Lecture 4: Two-sample t test
Example
Informal Analysis
Response: Blood pressure
(mmHg)
Treatments: Low sodium (50
mmol/day), High sodium (200
mmol/day).
Experimental material: 20 men
with Stage 1 hypertension.
Graph the data.
Visually compare the two sets
of data.
Level
Variability
1
2
Informal Analysis
The men in the 200 mmol/day
diet tend to have higher blood
pressures then those on the 50
mmol/day diet.
Both groups tend to have
about the same variability.
3
Informal Analysis
N
Mean
Standard
Deviation
50 mmol/day
Diet
10
143.1 mmHg
11.75 mmHg
4
Informal Analysis
On average, the 10 men on
the 50 mmol/day diet have
lower blood pressure by
19.9 mmHg.
200 mmol/day
Diet
10
163.0 mmHg
10.10 mmHg
5
6
1
Lecture 4: Two-sample t test
Informal Analysis
Formal Analysis
Blood pressures for men in
both diets show about the
same amount of variation.
Model:

ε
i = 50 or 200

random error
7
8
Conditions
Test of Hypothesis


Random errors are
identically and normally
distributed with mean 0 and
standard deviation, .
:
:
or
or
0
0
 Directional alternative because there is
prior information about lower sodium
being good for blood pressure.
9
Estimation


10
Estimation
= 143.1 is an estimate of

= 11.75 is an estimate
of .
= 10.10 is an estimate

of .
= 163.0 is an estimate of

= –19.9 is an
estimate of
11
12
2
Lecture 4: Two-sample t test
Pooled Estimate of
1
Pooled Estimate of
9 11.75
1
9 10.10
18
2
120.05
10.96
13
Test Statistic
1
1
19.9
4.90
Interpretation
The difference in sample
means (–19.9 mmHg) is 4.06
standard errors below zero.
This is a large standardized
difference.
19.9
0
10.96
14
1
10
1
10
4.06
15
16
P-value
P-value
The probability of observing a
value of the test statistic as, or
more, extreme than the one
calculated, given the null
hypothesis is true.
P-value = 0.0004
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18
3
Lecture 4: Two-sample t test
Interpretation
Conclusion
The P-value is so small
(smaller than 0.05) that we
reject the null hypothesis in
favor of the alternative.
The difference in sample mean
blood pressures for the two
diets is statistically significant,
it cannot be explained by
random error alone.
19
Description
20
Conclusion
Men on the low sodium diet
had lower average blood
pressure than those on the
high sodium diet.
The difference in sample
mean blood pressures is
statistically significant.
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22
Inference
Generalization
The population mean for
the 50 mmol/day diet is
lower than than for the 200
mmol/day diet.
Men with Stage 1 hypertension
on a 50 mmol/day Na diet will
have lower average blood
pressure than men on a 200
mmol/day Na diet.
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