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