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Nuttirudee Charoenruk
Hypothesis Testing for Two Populations
Link hypothesis testing and estimation
Hypothesis Estimation
Upper tailed test
H
1
Lower tailed test
H
1
Two-tailed test
A (1-
)100% lower confidence bound (LCB)
= point estimator -
test statistic
∝
×
(standard error of the estimator)
A (1-
)100% upper confidence bound (UCB)
= point estimator +
test statistic
∝
×
(standard error of the estimator)
A (1- ∝ )100% confidence interval
= point estimator ±
test statistic
∝ / 2
×
(standard error of the estimator)
Hypothesis testing
1. Hypothesis testing for the difference between two means: A paired sample
Condition
Two samples are dependent
The distribution of the difference in the dependent variable between two related
groups should be approximately normally distributed.
Null hypothesis
H
0
: μ
d
= d
0
Test statistic
t =
d − μ
d
s
d
/√ n
(If pair sample is large, use z-test)
where
d =
∑
d
i
n
and
s
d
❑
√
∑
( d
i
❑
d )
2
n − 1
Alternative
Hypothesis
Rejection/
critical region
H
0
Estimation
H
1
: μ
d
d
0
H
1
: μ
d
< d
0
H
1
: μ
d
≠ d
0
t cal
t α; n-
t cal
< -t α; n-
|t cal
| > t α/2; n-
d − t
∝ ;n − 1
s
d
√
n
d + t
∝;n − 1
s
d
√
n
d ±t
∝ / 2 ;n − 1
s
d
√ n
Nuttirudee Charoenruk
2. Hypothesis testing for the difference between two means: Independent samples
(large samples or small sample with known population variances)
Condition
The two samples are independent
Each of the two populations is normally distributed
Samples:
o Large samples (
n
1
and
n
2
For unknown population variances, use the sample variance ( s
2
as an estimator of population variance ( σ
2
o At least one sample is small but known population variances ( σ
1
2
and σ
2
2
Null hypothesis
H
0
: μ
1
− μ
2
= k
Test statistic
z =
(
´ x
1
−´ x
2
)
−( μ
1
− μ
2
σ
1
2
n
1
σ
2
2
n
2
or
z =
(
´ x
1
−´ x
2
)
−( μ
1
− μ
2
s
1
2
n
1
s
2
n
2
Alternative
Hypothesis
Rejection/
critical region
H
0
Estimation
H
1
: μ
1
− μ
2
k
H
1
: μ
1
− μ
2
< k
H
1
: μ
1
− μ
2
≠ k
z cal
z α
z cal
< -z α
|z cal
| > z α/
(
´ x
1
−´ x
2
)
− z
α
σ
1
2
n
1
σ
2
2
n
2
or
(
´ x
1
−´ x
2
)
− z
α
s
1
2
n
1
s
2
2
n
2
(
´ x
1
−´ x
2
)
α
σ
1
2
n
1
σ
2
2
n
2
or
(
´ x
1
−´ x
2
)
α
s
1
2
n
1
s
2
2
n
2
(
´ x
1
−´ x
2
)
± z
α / 2
σ
1
2
n
1
σ
2
2
n
2
or
(
´ x
1
−´ x
2
)
± z
α / 2
s
1
2
n
1
s
2
2
n
2
3. Hypothesis testing for the difference between two means: Independent samples (small
samples with unknown population variances but assumed to be equal)
Condition
The two samples are independent
Each of the two populations is normally distributed
At least one sample is small with unknown population variances ( σ
1
2
and σ
2
2
) but
assumed to be equal
Nuttirudee Charoenruk
H
1
: μ
1
− μ
2
k
H
1
: μ
1
− μ
2
< k
H
1
: μ
1
− μ
2
≠ k
t
cal
t
α ;df
t cal
t
α ;df
|t
cal
t
α
2
; df
❑
(
´ x
1
−´ x
2
)
− t
α ;df
s
1
2
n
1
s
2
2
n
2
(
´ x
1
−´ x
2
)
α; df
s
1
2
n
1
s
2
2
n
2
(
´ x
1
−´ x
2
)
±t
α
2
;df
s
1
2
n
1
s
2
2
n
2
Nuttirudee Charoenruk
5. Hypothesis testing for the equality of population variance
Null hypothesis H
0
: σ
1
2
= σ
2
2
σ
1
2
σ
2
2
Test statistic
F =
s
i
2
s
j
2
if s
i
2
> s
j
2
; i, j = 1, 2
Reject H 0
if F > F α/
at d.f. n i
-1 and n j
6. Hypothesis testing for the difference between two proportions: Test equal proportions
Condition
The two samples are independent
n
1
^ p ≥ 5 ,n
1
q ^≥ 5 , n
2
p ^≥ 5 and n
2
q ^≥ 5 ,
Null hypothesis
H
0
: p
1
− p
2
Test statistic
z =
(
^ p
1
− ^ p
2
)
−( p
1
− p
2
√
^ p q ^(
n
1
n
2
^
p =
x
1
2
n
1
2
and
^
q = 1 −
^
p
Alternative
Hypothesis
Rejection/
critical region
H
0
Estimation
H
1
: p
1
− p
2
H
1
: p
1
− p
2
H
1
: p
1
− p
2
z cal
z
α
z
cal
z
α
|z cal
z
∝ / 2
(
^
p
1
^
p
2
)
− z
α
√
^
p
^
q (
n
1
n
2
(
^
p
1
^
p
2
)
α
√
^
p
^
q (
n
1
n
2
(
^
p
1
^
p
2
) ± z
α / 2
√
^
p
^
q (
n
1
n
2
