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

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

UpStudy AI · Accepted Answer

We need to check the condition for using the normal approximation for the distribution of the sample proportion. A common requirement is that both

$$
np \ge 10 \quad \text{and} \quad n(1-p) \ge 10,
$$

where $$ p $$ is the population proportion. Here, $$ p = 0.005 $$.

### Step 1: Checking $$ np \ge 10 $$

The condition becomes:

$$
n \cdot 0.005 \ge 10 \quad \Rightarrow \quad n \ge \frac{10}{0.005} = 2000.
$$

So, the sample size must be at least $$ 2000 $$ for $$ np $$ to be at least $$ 10 $$.

### Step 2: Checking $$ n(1-p) \ge 10 $$

Since $$ 1-p = 0.995 $$, for any $$ n \ge 2000 $$:

$$
n \cdot 0.995 \ge 2000 \cdot 0.995 = 1990,
$$

which is much greater than $$ 10 $$. Therefore, the condition $$ n(1-p) \ge 10 $$ is automatically satisfied for these choices of $$ n $$.

### Step 3: Evaluating Given Sample Sizes

We now check each sample size:

- **2600:** Since $$ 2600 \ge 2000 $$, it is sufficient.
- **1800:** Since $$ 1800 < 2000 $$, it is not sufficient.
- **4500:** Since $$ 4500 \ge 2000 $$, it is sufficient.
- **3700:** Since $$ 3700 \ge 2000 $$, it is sufficient.
- **1900:** Since $$ 1900 < 2000 $$, it is not sufficient.

### Final Answer

The sample sizes that are sufficiently large are:

- $$2600$$
- $$4500$$
- $$3700$$
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.

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