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.

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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: $$ 0.0110 $$
(a) The mean is 0.038. (b) The standard deviation is 0.0110.

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