The theorem that states that the sampling distribution of the sample mean is approximately normal when the sample is large is called the

The theorem you're referring to is the Central Limit Theorem (CLT).

According to the Central Limit Theorem, if the population has a mean (\(\mu\)) of 200 and standard deviation (\(\sigma\)) of 25, then the sampling distribution of the sample mean (\(\bar{x}\)) for a sample size of \(n = 100\) has:

- Mean (\(\mu_{\bar{x}}\)): This is equal to the population mean, so \(\mu_{\bar{x}} = 200\).

- Standard Deviation (\(\sigma_{\bar{x}}\)): This is calculated using the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\). Given that \(\sigma = 25\) and \(n = 100\):

\[
\sigma_{\bar{x}} = \frac{25}{\sqrt{100}} = \frac{25}{10} = 2.5.
\]

Therefore, the answers are:
- Mean: 200
- Standard Deviation: 2.5