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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UpStudy AI · Accepted Answer

**Step 1.**  
The population proportion is given as  
$$
p = 0.385
$$  
and the sample size is  
$$
n = 40.
$$

**Step 2.**  
The standard deviation of the sampling distribution for sample proportions (often also called the standard error) is calculated by  
$$
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}.
$$

**Step 3.**  
Substitute the values into the formula:  
$$
\sigma_{\hat{p}} = \sqrt{\frac{0.385 \times (1-0.385)}{40}} = \sqrt{\frac{0.385 \times 0.615}{40}}.
$$

**Step 4.**  
Multiply $$0.385$$ and $$0.615$$:  
$$
0.385 \times 0.615 \approx 0.2368.
$$

**Step 5.**  
Divide by $$40$$:  
$$
\frac{0.2368}{40} \approx 0.00592.
$$

**Step 6.**  
Take the square root:  
$$
\sigma_{\hat{p}} \approx \sqrt{0.00592} \approx 0.0770.
$$

**(a)**  
$$
\sigma_{\hat{p}} \approx 0.0770.
$$

**Step 7.**  
The standard error (SE) of the sampling distribution for sample proportions is calculated with the same formula, hence  
$$
SE \approx 0.0770.
$$

**(b)**  
$$
SE \approx 0.0770.
$$
**(a)** $$ \sigma_{\hat{p}} \approx 0.0770 $$ **(b)** $$ SE \approx 0.0770 $$

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