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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To determine if a sample size is sufficiently large for using a normal distribution, we can apply the rule of thumb that both \( np \) and \( n(1-p) \) should be greater than 5, where \( p \) is the proportion of the population (0.005 for 0.5%). Calculating for the sample sizes given: - For 1900: \( np = 1900 \times 0.005 = 9.5 \) and \( n(1-p) = 1900 \times 0.995 = 1885 \) (sufficient) - For 2200: \( np = 2200 \times 0.005 = 11 \) and \( n(1-p) = 2200 \times 0.995 = 2189 \) (sufficient) - For 4100: \( np = 4100 \times 0.005 = 20.5 \) and \( n(1-p) = 4100 \times 0.995 = 4079.5 \) (sufficient) - For 4300: \( np = 4300 \times 0.005 = 21.5 \) and \( n(1-p) = 4300 \times 0.995 = 4278.5 \) (sufficient) - For 2100: \( np = 2100 \times 0.005 = 10.5 \) and \( n(1-p) = 2100 \times 0.995 = 2089.5 \) (sufficient) All given sample sizes (1900, 2200, 4100, 4300, and 2100) are sufficiently large for using a normal distribution in this context.
