Introduction:
The random error in the arithmetic mean of a sample of observations is a measure of the variability of the sample mean. As the sample size increases, the random error in the sample mean decreases. In this question, we will determine the random error in the arithmetic mean of 150 observations given the random error in the arithmetic mean of 50 observations.

Calculation:
The standard error of the mean (SEM) is a measure of the variability of the sample mean. It is given by the formula:
SEM = s / sqrt(n)
where s is the standard deviation of the sample and n is the sample size.

If the random error in the arithmetic mean of 50 observations is alpha, then the SEM of the sample can be calculated as:
alpha = s / sqrt(50)
s = alpha * sqrt(50)

To calculate the random error in the arithmetic mean of 150 observations, we need to calculate the SEM of the sample with 150 observations. Let's call this random error beta. We can calculate beta as:
beta = s / sqrt(150)
beta = (alpha * sqrt(50)) / sqrt(150)

Therefore, the random error in the arithmetic mean of 150 observations is:
beta = (alpha * sqrt(50)) / sqrt(150)

Conclusion:
As the sample size increases, the random error in the arithmetic mean decreases. In this question, we have shown that the random error in the arithmetic mean of 150 observations can be calculated from the random error in the arithmetic mean of 50 observations using the formula beta = (alpha * sqrt(50)) / sqrt(150).