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.
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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.