Stat 312: Lecture 16
Other two sample tests
Moo K. Chung
[email protected]
March 25, 2003
Concepts
1. Paired
data:
for
a
given
paired
sample
(
X
1
, Y
1
)
,
· · ·
,
(
X
n
, Y
n
)
with
X
i
=
μ
X
and
Y
i
=
μ
Y
, a test statistic for testing
H
0
:
μ
X
=
μ
Y
can
be based on one sample test.
2. Let
X
i
∼
Bernoulli
(
p
X
)
and
Y
j
∼
Bernoulli
(
p
Y
)
. Let
ˆ
p
X
=
∑
n
i
=1
X
i
/n
and
ˆ
p
Y
=
∑
m
j
=1
Y
j
/m
.
Var
(ˆ
p
X

ˆ
p
Y
) =
p
X
(1

p
X
)
n
+
p
Y
(1

p
Y
)
m
.
3. Difference between population proportions: for large
n
and
m
, use a
Z
statistic for testing
H
0
:
p
X
=
p
Y
:
Z
=
ˆ
p
X

ˆ
p
Y

(ˆ
p
X

ˆ
p
Y
)
p
Var
(ˆ
p
X

ˆ
p
Y
)
∼
N
(0
,
1)
.
Since
p
X
and
p
Y
are unknown, we estimate them from
the samples.
Inclass problems
Example 1.
10 students took two midterm exams.
Student
k
01 02 03 04 05 06 07 08 09 10
Midterm 1
k
80 75 60 90 99 60 55 85 65 70
Midterm 2
k
70 60 70 72 95 66 60 80 70 60
Is the first exam easier than the second exam? Test it at level
0.05.
Solution.
Let
X
i
and
Y
i
be the first and the second midterm
scores for the
i
th student. Perform the
t
test on
H
0
:
μ
Y

μ
X
= 0
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 Spring '04
 Chung
 Statistics, Normal Distribution, Harshad number, Statistical hypothesis testing, H0

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