C

Step 1: The random error in the arithmetic mean (often referred to as the standard error) is calculated using the formula: \(\text{SE} = \frac{s}{\sqrt{n}}\), where \(s\) is the standard deviation and \(n\) is the number of observations.

Step 2: For 100 observations, the random error is given as \(x\). Hence, we have: \(x = \frac{s}{\sqrt{100}} = \frac{s}{10}\).

Step 3: For 400 observations, let the random error be \(y\). Therefore, we have: \(y = \frac{s}{\sqrt{400}} = \frac{s}{20}\).

Step 4: To express \(y\) in terms of \(x\), we can relate them as follows: \(y = \frac{s}{20} = \frac{s}{10} \cdot \frac{1}{2} = \frac{x}{2}\).

Step 5: This indicates that the random error in the arithmetic mean of 400 observations is half of that of 100 observations, hence, \(y = 2x\).

Therefore, the correct answer is (C) 2x.