D

The random error in the arithmetic mean (also known as the standard error of the mean) is inversely related to the square root of the sample size, given by the formula:
\[E = \frac{\sigma}{\sqrt{n}}\]
For 100 observations, let the error be \(x\):
\[x = \frac{\sigma}{\sqrt{100}} = \frac{\sigma}{10}\]
Now for 400 observations:
\[E' = \frac{\sigma}{\sqrt{400}} = \frac{\sigma}{20}\]
We can express \(E'\) in terms of \(x\):
\[E' = \frac{\sigma}{20} = \frac{1}{2} \cdot \frac{\sigma}{10} = \frac{1}{2} x\]
Thus, the random error in the arithmetic mean of 400 observations is \( \frac{1}{2} x \).
Therefore, the correct answer is D: \( \frac{1}{2} x \).