Transcript Unpaired t-test

Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples:
- We have samples for each of two conditions.
We provide an answer for
“Are the two sample means significantly different from
each other, or could both plausibly come from the
same population?”
Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples:
H 0 : 1  2 , 1  2  0
H a : 1   2 (two tailed test)
1   2 or 1   2 (one - tailed test)
Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples:
Sample 1
Sample 2
Sample size:
n1
n2
Sample mean:
x1
x2
s1
s2
Sample
standard
deviation:
Combined
estimate of
variance:
s1 (n1  1)  s 2 (n2  1)

(n1  1)  (n2  1)
2
sc
Variance of the
difference between
the two means:
2
2
2
s
2
( x1  x 2 )
2
Pooled or
combined variance
sc
sc
1
2 1


 sc (  )
n1
n2
n1 n2
S1 and S2 must be compatible.
Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples:
Combined
estimate of
variance:
s1 (n1  1)  s 2 (n2  1)

(n1  1)  (n2  1)
2
sc
2
2
S1 and S2 must be compatible:
- If the variances are not significantly different at
the 10% level of significance, they can be combined
to give a better estimate of variance to use in the t-test.
(p.255, DeCoursey)
- Larger departures from equality of the variances can be
tolerated if the two samples are of equal size.
n1=n2
(p.234, DeCoursey)
Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples
The test statistics t is
x1  x 2
t
s( x  x )
1
or
2
2
s
2
( x1  x 2 )
y
t
sy
2
sc
sc
1
2 1
 sy 

 sc (  )
n1
n2
n1 n2
2
When comparing the sample
mean to a population mean:
Normal Distribution:
x
t
s
( )
n
z
x
(

n
)
Statistical Inference for the Mean: t-test
Comparison of sample means using unpaired samples
The degrees of freedom are
s1 (n1  1)  s 2 (n2  1)

(n1  1)  (n2  1)
2
df = n1-1+n2-1
sc
2
When comparing the sample mean to
a population mean: n (sample size)
df  n  1
Normal Distribution: no such a parameter.
2
Statistical Inference for the Mean: t-test
Test of Significance: Comparing the sample means
using unpaired samples
- State the null hypothesis in terms of means, such as 1   2 or  y  0.
- State the alternative hypothesis in terms of the same population
parameters.
- Determine the combined variance and the variance of the
difference of the sample means.
- Calculate the test statistic t of the observation using the means and
variance.
x1  x 2
df
=
n
-1+n
-1
t
- Determine the degrees of freedom:
1
2
s( x  x )
- State the level of significance – rejection limit.
1
2
- If probability falls outside of the rejection limit, we reject the Null
Hypothesis, which means the sample mean is significantly different
from the other one.
Assume the sample means are normally distributed.