This is a test of 2 population proportions.
The problem asks for a difference in proportions.
Let AA and BB be the subscripts for medication A and medication B. Then
p
A
p
A
and
p
B
p
B
are the desired population proportions.
P'
A
−
P'B
=
P'
A
−
P'B
= difference in the proportions of adult patients who did not
react after 30 minutes to medication A and medication B.
H
o
:
p
A
=
p
B
p
A
-
p
B
=
0
H
o
:
p
A
=
p
B
p
A
-
p
B
=0
H
a
:
p
A
≠
p
B
p
A
-
p
B
≠
0
H
a
:
p
A
≠
p
B
p
A
-
p
B
≠0
The words "is a difference"
tell you the test is two-tailed.
Distribution for the test: Since this is a test of two binomial population proportions,
the distribution is normal:
p
c
=
x
A
+
x
B
n
A
+
n
B
=
20
+
12
200
+
200
=
0.08 1
−
p
c
=
0.92
p
c
=
x
A
+
x
B
n
A
+
n
B
=
20
+
12
200
+
200
=0.081−
p
c
=0.92
Therefore,
P'
A
−
P'B
~
N
[
0
,
(
0.08
)
⋅
(
0.92
)
⋅
(
1
200
+
1
200
)
]
P'
A
−
P'B
~N[0,
(
0.08
)
⋅
(
0.92
)
⋅
(
1
200
+
1
200
)
]
P'
A
−
P'B
P'
A
−
P'B
follows an approximate normal distribution.
Calculate the p-value using the normal distribution: p-value = 0.1404.
Estimated proportion for group A:
p'
A
=
x
A
n
A
=
20
200
=
0.1
p'
A
=
x
A
n
A
=
20
200
=0.1
Estimated proportion for group B:
p'
B
=
x
B
n
B
=
12
200
=
0.06
p'
B
=
x
B
n
B
=
12
200
=0.06
P'
A
−
P'B
=
0.1
-
0.06
=
0.04
P'
A
−
P'B
=0.1-0.06=0.04.
Half the p-value is
below -0.04 and
half is above 0.04.
Compare αα and the p-value: α=0.01α=0.01 and the p-value = 0.1404p-value= 0.1404.
α
<
α< p-value.
Make a decision: Since
α
<
p-value
α<p-value, do not reject
H
o
H
o
.
Conclusion: At a 1% level of significance, from the sample data, there is not
sufficient evidence to conclude that there is a difference in the proportions of adult
patients who did not react after 30 minutes to medication A and medication B.
TI-83+ and TI-84: Press STAT. Arrow over to TESTS and press
6:2-PropZTest. Arrow down and enter 20 for x1x1, 200 for n1n1, 12 for x2x2,
and 200 for n2n2. Arrow down to p1: and arrow to not equal p2. Press
ENTER. Arrow down to Calculate and press ENTER. The p-value is p =
0.1404p =0.1404 and the test statistic is 1.47. Do the procedure again but instead
of Calculate do Draw.
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