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 2600 1800 4500 3700 1900

To determine if a sample size is sufficiently large for estimating proportions using a normal distribution, the rule of thumb is that both \( np \) and \( n(1-p) \) should be greater than or equal to 10, where \( n \) is the sample size and \( p \) is the population proportion. In this case, \( p = 0.005 \) (0.5%) and \( 1-p = 0.995 \). Calculating for each of the sample sizes: - For \( n = 2600 \): - \( np = 2600 \times 0.005 = 13 \) (sufficient) - \( n(1-p) = 2600 \times 0.995 = 2587 \) (sufficient) - For \( n = 1800 \): - \( np = 1800 \times 0.005 = 9 \) (not sufficient) - \( n(1-p) = 1800 \times 0.995 = 1791 \) (sufficient) - For \( n = 4500 \): - \( np = 4500 \times 0.005 = 22.5 \) (sufficient) - \( n(1-p) = 4500 \times 0.995 = 4477.5 \) (sufficient) - For \( n = 3700 \): - \( np = 3700 \times 0.005 = 18.5 \) (sufficient) - \( n(1-p) = 3700 \times 0.995 = 3685.5 \) (sufficient) - For \( n = 1900 \): - \( np = 1900 \times 0.005 = 9.5 \) (not sufficient) - \( n(1-p) = 1900 \times 0.995 = 1890.5 \) (sufficient) Thus, the sample sizes that are sufficiently large for this situation are **2600**, **4500**, and **3700**.