If observation is increased by 'n' times, then random error decreases by =1/n
Therefore, random error of 150 observations = x/ 3
Thus, option c) x/3 is correct.

Explanation:

The standard error of the mean(SEM) indicates how far the sample mean is likely to be from the true population mean.

Formula:

The standard error of the mean(SEM) is given by:
SEM= s/√n
where s is the standard deviation of the sample and n is the sample size.

Calculation:

Let's assume that the standard deviation(s) of the 50 observations is x1, and the standard deviation(s) of the 150 observations is x2.
Then, the SEM for 50 observations is given by:
SEM1= x1/√50

Similarly, the SEM for 150 observations is given by:
SEM2= x2/√150

Now, we need to find the relationship between SEM1 and SEM2.

Solution:

We know that the standard deviation of a larger sample is expected to be smaller than the standard deviation of a smaller sample.

Therefore, we can assume that x2 is smaller than x1.

We also know that the standard error of the mean(SEM) is inversely proportional to the square root of the sample size(n).

Therefore, we can say that:
SEM2/SEM1=√(50/150)
SEM2/SEM1=1/√3

Hence, the random error in the arithmetic mean of 150 observations would be X/√3, which is approximately equal to 0.58x.

Therefore, the correct option is 3) X/3.

Conclusion:

The random error in the arithmetic mean of a larger sample is expected to be smaller than the random error in the arithmetic mean of a smaller sample, due to the inverse relationship between SEM and sample size. Hence, we can use the formula SEM= s/√n to calculate the SEM for different sample sizes, and determine the relationship between them.