if the randon error in the arithmetic mean of 50 observations is `alpha`, then the random error in the arithmetic mean of 150 observations would be

Options

A. `alpha`
B. `3alpha`
C. `(alpha)/(3)`
D. `2alpha`

Answer

To solve the problem, we need to understand how the random error in the arithmetic mean changes with the number of observations.

### Step-by-Step Solution:

1. **Understanding Random Error**: The random error in the arithmetic mean is inversely proportional to the square root of the number of observations. This means that as the number of observations increases, the random error decreases.

2. **Given Information**: We are given that the random error in the arithmetic mean of 50 observations is denoted as `alpha`.

3. **Formula for Random Error**: The random error (E) in the arithmetic mean can be expressed as:
\[ E = \frac{\sigma}{\sqrt{n}} \]
where \( \sigma \) is the standard deviation of the observations and \( n \) is the number of observations.

4. **Random Error for 50 Observations**: For 50 observations, the random error is:
\[ E_{50} = \frac{\sigma}{\sqrt{50}} = \alpha \]

5. **Finding Random Error for 150 Observations**: Now, we want to find the random error for 150 observations:
\[ E_{150} = \frac{\sigma}{\sqrt{150}} \]

6. **Relating the Two Errors**: We can relate the two errors:
\[ E_{150} = E_{50} \cdot \frac{\sqrt{50}}{\sqrt{150}} \]
Substituting \( E_{50} = \alpha \):
\[ E_{150} = \alpha \cdot \frac{\sqrt{50}}{\sqrt{150}} \]

7. **Simplifying the Expression**: We know that:
\[ \frac{\sqrt{50}}{\sqrt{150}} = \frac{\sqrt{50}}{\sqrt{50 \cdot 3}} = \frac{1}{\sqrt{3}} \]
Therefore:
\[ E_{150} = \alpha \cdot \frac{1}{\sqrt{3}} = \frac{\alpha}{\sqrt{3}} \]

8. **Final Result**: Thus, the random error in the arithmetic mean of 150 observations is:
\[ E_{150} = \frac{\alpha}{\sqrt{3}} \]