According to the UCLA School of Law Williams Institute, of the 1.3 million adults who identify as transgender, \( 38.5 \% \) are transgender women. In a separate study, researchers collected a random sample of 40 adults who identify as transgender and found \( 46.2 \% \) were transgender women. (a) Calculate the standard deviation of the sampling distribution for sample proportions if repeated samples of size \( n=40 \) are taken. \( \sigma_{\hat{p}}=\square \) Round your answer to four decimal places. (b) Calculate, the standard error of the sampling distribution for sample proportions with a sample size of 40 . \( S E=\square \) Round your answer to four decimal places.

To calculate the standard deviation and the standard error of the sampling distribution for sample proportions, we can use the following formulas: (a) The standard deviation of the sampling distribution for sample proportions can be calculated using the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion (0.385 for transgender women) and \( n \) is the sample size (40). Substituting the values: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.385(1-0.385)}{40}} = \sqrt{\frac{0.385 \times 0.615}{40}} = \sqrt{\frac{0.236775}{40}} = \sqrt{0.005919375} \approx 0.0770 \] So, \( \sigma_{\hat{p}} \approx 0.0770 \). (b) The standard error of the sampling distribution for sample proportions is simply the standard deviation calculated above. Thus, it remains the same: \[ SE = \sigma_{\hat{p}} = 0.0770 \] So, \( SE \approx 0.0770 \) as well.