According to the National Institute of Mental Health the prevalence of a current ADHD diagnosis amongst U.S. Adults aged \( 25-34 \) from 2001-2003 was \( 3.8 \% \). Calculate the mean and standard deviation for a sampling distribution using sample size 300 . (a) The mean is Do not round this answer. (b) The standard deviation is Round this answer to 4 decimal places.

To calculate the mean and standard deviation for a sampling distribution of the proportion of adults diagnosed with ADHD, we can use the following formulas: 1. **Mean of the sampling distribution (\( \mu \))**: \[ \mu = p \] where \( p \) is the population proportion. 2. **Standard deviation of the sampling distribution (\( \sigma \))**: \[ \sigma = \sqrt{\frac{p(1 - p)}{n}} \] where \( n \) is the sample size. Given: - \( p = 0.038 \) (which is \( 3.8\% \)) - \( n = 300 \) Now, let's calculate the mean and standard deviation step by step. ### Step 1: Calculate the Mean The mean of the sampling distribution is simply the population proportion: \[ \mu = p = 0.038 \] ### Step 2: Calculate the Standard Deviation Now, we will calculate the standard deviation using the formula: \[ \sigma = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.038(1 - 0.038)}{300}} \] Let's perform the calculation for the standard deviation. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(0.038\left(1-0.038\right)\right)}{300}}\) - step1: Remove the parentheses: \(\sqrt{\frac{0.038\left(1-0.038\right)}{300}}\) - step2: Subtract the numbers: \(\sqrt{\frac{0.038\times 0.962}{300}}\) - step3: Multiply the numbers: \(\sqrt{\frac{0.036556}{300}}\) - step4: Divide the terms: \(\sqrt{\frac{9139}{75000000}}\) - step5: Use the properties of radicals: \(\frac{\sqrt{9139}}{\sqrt{75000000}}\) - step6: Simplify the expression: \(\frac{\sqrt{9139}}{5000\sqrt{3}}\) - step7: Multiply by the Conjugate: \(\frac{\sqrt{9139}\times \sqrt{3}}{5000\sqrt{3}\times \sqrt{3}}\) - step8: Multiply the numbers: \(\frac{\sqrt{27417}}{5000\sqrt{3}\times \sqrt{3}}\) - step9: Multiply the numbers: \(\frac{\sqrt{27417}}{15000}\) Calculate or simplify the expression \( 0.011039 \). Numbers by following steps: - step0: Solution: \(0.011039\) Now that we have performed the calculations, here are the results: (a) The mean of the sampling distribution is: \[ \mu = 0.038 \] (b) The standard deviation of the sampling distribution, rounded to four decimal places, is: \[ \sigma \approx 0.0110 \] Thus, the final answers are: - Mean: \( 0.038 \) - Standard Deviation: \( 0.0110 \)