Question
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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

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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Simplify this solution

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.

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