When 10 observation are taken the random error is x when 100 observation are taken the random error becomes

Options

A. `x//10`
B. `x^2`
C. `10x`
D. `sqrt(x)`

Answer

To solve the problem, we need to understand the relationship between the number of observations and the random error. Here’s a step-by-step solution:

### Step 1: Understand the relationship
Random error (r) is inversely proportional to the number of observations (n). This means:
\[ r \propto \frac{1}{n} \]
This can be expressed as:
\[ r = \frac{k}{n} \]
where \( k \) is a constant.

### Step 2: Write the equation for the first scenario
For the first case with 10 observations:
\[ r_1 = \frac{k}{10} \]
Given that \( r_1 = x \), we can write:
\[ x = \frac{k}{10} \]

### Step 3: Write the equation for the second scenario
For the second case with 100 observations:
\[ r_2 = \frac{k}{100} \]

### Step 4: Relate the two random errors
Now, we can relate \( r_1 \) and \( r_2 \):
\[ \frac{r_1}{r_2} = \frac{\frac{k}{10}}{\frac{k}{100}} \]
This simplifies to:
\[ \frac{r_1}{r_2} = \frac{100}{10} = 10 \]

### Step 5: Substitute \( r_1 \) with \( x \)
Since \( r_1 = x \):
\[ \frac{x}{r_2} = 10 \]

### Step 6: Solve for \( r_2 \)
Rearranging the equation gives:
\[ r_2 = \frac{x}{10} \]

### Conclusion
Thus, when 100 observations are taken, the random error becomes:
\[ r_2 = \frac{x}{10} \]

### Final Answer
The random error when 100 observations are taken is \( \frac{x}{10} \).

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