Answer: 1/2
Step-by-step explanation:
This is straight from Khan
ample size and margin of error
In a one-sample
z
zz interval for a proportion, the
margin of error
margin of errorstart text, start color #11accd, m, a, r, g, i, n, space, o, f, space, e, r, r, o, r, end color #11accd, end text is how far our interval extends above and below the sample proportion:
(
statistic
)
±
(
margin of error
)
p
^
±
z
∗
p
^
(
1
−
p
^
)
n
(statistic)
p
^
±(margin of error)
±z
∗
n
p
^
(1−
p
^
)
Since the sample size
n
nn is in the denominator, increasing the sample size will lead to a smaller margin of error.
Hint #22 / 3
Comparing sizes of margins of error
Let's look at each margin of error in terms of its sample size.
Smaller sample,
n
=
200
n=200n, equals, 200:
=
z
∗
p
^
(
1
−
p
^
)
200
error
margin of
=z
∗
200
p
^
(1−
p
^
)
start text, e, r, r, o, r, end text, start superscript, start text, m, a, r, g, i, n, space, o, f, end text, end superscript, equals, z, start superscript, times, end superscript, square root of, start fraction, p, with, hat, on top, left parenthesis, 1, minus, p, with, hat, on top, right parenthesis, divided by, 200, end fraction, end square root
Larger sample,
n
=
800
n=800n, equals, 800:
=
z
∗
p
^
(
1
−
p
^
)
800
=
z
∗
p
^
(
1
−
p
^
)
4
⋅
200
=
z
∗
1
4
⋅
p
^
(
1
−
p
^
)
200
=
1
2
⋅
z
∗
p
^
(
1
−
p
^
)
200
error
margin of
=z
∗
800
p
^
(1−
p
^
)
=z
∗
4⋅200
p
^
(1−
p
^
)
=z
∗
4
1
⋅
200
p
^
(1−
p
^
)
=
2
1
⋅z
∗
200
p
^
(1−
p
^
)
Even though
800
800800 is
4
44 times larger than the smaller sample, the margin of error from the larger sample is
1
2
2
1
start fraction, 1, divided by, 2, end fraction the margin of error from the smaller sample.
Equivalently, the margin of error from the smaller sample is
2
22 times the margin of error from the larger sample.
Hint #33 / 3
Answer
The margin of error from the larger sample will be
1
2
2
1
start fraction, 1, divided by, 2, end fraction the margin of error from the smaller sample.
