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 3700 4400 4600 500 4200

To determine whether a sample size is sufficiently large when estimating proportions using a normal distribution, we can use the rule of thumb that both \( np \) and \( n(1-p) \) should be greater than 5, where \( p \) is the proportion being estimated. In this case, \( p = 0.005 \) (0.5%). - For \( n = 3700 \): \( np = 3700 \times 0.005 = 18.5 \) and \( n(1-p) = 3700 \times 0.995 \approx 3681.5 \) -> Sufficient. - For \( n = 4400 \): \( np = 4400 \times 0.005 = 22 \) and \( n(1-p) = 4400 \times 0.995 \approx 4382 \) -> Sufficient. - For \( n = 4600 \): \( np = 4600 \times 0.005 = 23 \) and \( n(1-p) = 4600 \times 0.995 \approx 4573 \) -> Sufficient. - For \( n = 500 \): \( np = 500 \times 0.005 = 2.5 \) and \( n(1-p) = 500 \times 0.995 \approx 497.5 \) -> Not sufficient. - For \( n = 4200 \): \( np = 4200 \times 0.005 = 21 \) and \( n(1-p) = 4200 \times 0.995 \approx 4179 \) -> Sufficient. Thus, the sample sizes of 3700, 4400, 4600, and 4200 are sufficiently large.