Question
The UCLA School of Law Williams Institute estimates that
of U.S. identify as transgender. If the researchers at the
Williams Institute want to use a normal distribution to estimate probabilities for the proportion of adults that identify as
transgender, which of the following sample sizes are sufficiently large?
Select all that apply
2600
1800
4500
3700
1900
Williams Institute want to use a normal distribution to estimate probabilities for the proportion of adults that identify as
transgender, which of the following sample sizes are sufficiently large?
Select all that apply
2600
1800
4500
3700
1900
Ask by Schwartz Howell. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The sample sizes of 2600, 4500, and 3700 are sufficiently large for using a normal distribution to estimate the proportion of adults identifying as transgender.
Solution
We need to check the condition for using the normal approximation for the distribution of the sample proportion. A common requirement is that both
where
is the population proportion. Here,
.
Step 1: Checking
The condition becomes:
So, the sample size must be at least
for
to be at least
.
Step 2: Checking
Since
, for any
:
which is much greater than
. Therefore, the condition
is automatically satisfied for these choices of
.
Step 3: Evaluating Given Sample Sizes
We now check each sample size:
- 2600: Since
, it is sufficient. - 1800: Since
, it is not sufficient. - 4500: Since
, it is sufficient. - 3700: Since
, it is sufficient. - 1900: Since
, it is not sufficient.
Final Answer
The sample sizes that are sufficiently large are:
Answered by UpStudy AI and reviewed by a Professional Tutor
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The Deep Dive
To determine if a sample size is sufficiently large for estimating proportions using a normal distribution, the rule of thumb is that both
and
should be greater than or equal to 10, where
is the sample size and
is the population proportion.
In this case,
(0.5%) and
.
Calculating for each of the sample sizes:
-
For
: -
(sufficient) -
(sufficient)
-
-
For
: -
(not sufficient) -
(sufficient)
-
-
For
: -
(sufficient) -
(sufficient)
-
-
For
: -
(sufficient) -
(sufficient)
-
-
For
: -
(not sufficient) -
(sufficient)
-
Thus, the sample sizes that are sufficiently large for this situation are 2600, 4500, and 3700.