The UCLA School of Law Williams Institute estimates that $$ 0.5 \% $$ 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 1900 2200 4100 4300 2100

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UpStudy AI · Accepted Answer

We are given that $$ p = 0.005 $$ (which corresponds to $$ 0.5\% $$) and must check the conditions for a normal approximation to the binomial proportion $$ \hat{p} $$ in a sample of size $$ n $$. A common rule is to require both

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

Since $$ p $$ is very small, $$ np $$ is the more restrictive condition. We compute $$ np $$ for each sample size:

- For $$ n = 1900 $$:
  $$
  1900 \times 0.005 = 9.5.
  $$
  Since $$ 9.5 < 10 $$, this sample size is not sufficiently large.

- For $$ n = 2200 $$:
  $$
  2200 \times 0.005 = 11.
  $$
  Since $$ 11 \ge 10 $$, this sample size is sufficiently large.

- For $$ n = 4100 $$:
  $$
  4100 \times 0.005 = 20.5.
  $$
  Since $$ 20.5 \ge 10 $$, this sample size is sufficiently large.

- For $$ n = 4300 $$:
  $$
  4300 \times 0.005 = 21.5.
  $$
  Since $$ 21.5 \ge 10 $$, this sample size is sufficiently large.

- For $$ n = 2100 $$:
  $$
  2100 \times 0.005 = 10.5.
  $$
  Since $$ 10.5 \ge 10 $$, this sample size is sufficiently large.

Thus, the sample sizes that are sufficiently large are:

- $$ 2200 $$
- $$ 4100 $$
- $$ 4300 $$
- $$ 2100 $$
The sample sizes of 2200, 4100, 4300, and 2100 are sufficiently large for using a normal distribution to estimate the proportion of adults identifying as transgender.

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