The geometric mean of 6 observation was calculated as 13. It was later observed that one of the observation was recorded as 28 instead of 36. The correct geometric mean is

Options

A. `((9)/(7))^((1)/(6))`
B. `3((9)/(7))^((1)/(6))`
C. `13((9)/(7))^((1)/(6))`
D. `13((7)/(9))^((1)/(6))`

Answer

To find the correct geometric mean after correcting the observation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given information**:
- The geometric mean (GM) of 6 observations was initially calculated as 13.
- One observation was recorded as 28 instead of the correct value of 36.

2. **Set up the geometric mean formula**:
The formula for the geometric mean of \( n \) observations \( A_1, A_2, \ldots, A_n \) is given by:
\[ GM = (A_1 \times A_2 \times \ldots \times A_n)^{\frac{1}{n}} \]
In this case, \( n = 6 \).

3. **Calculate the product of the observations**:
Since the GM was calculated as 13, we can express the product of the observations as:
\[ A_1 \times A_2 \times A_3 \times A_4 \times A_5 \times A_6 = 13^6 \]

4. **Identify the incorrect observation**:
Let’s denote the product of the other observations (excluding the incorrect one) as \( A \):
\[ A_1 = 28 \quad \text{(incorrect)} \]
Therefore, the product can be expressed as:
\[ 28 \times A = 13^6 \]

5. **Calculate the correct product with the correct observation**:
Now, replace the incorrect observation with the correct one:
\[ A_1 = 36 \quad \text{(correct)} \]
The new product of the observations becomes:
\[ 36 \times A \]

6. **Set up the equation for the new geometric mean**:
The new geometric mean can be calculated as:
\[ GM' = (36 \times A)^{\frac{1}{6}} \]
Since \( A = \frac{13^6}{28} \), we substitute this into the equation:
\[ GM' = \left(36 \times \frac{13^6}{28}\right)^{\frac{1}{6}} \]

7. **Simplify the expression**:
\[ GM' = \left(\frac{36 \times 13^6}{28}\right)^{\frac{1}{6}} = \left(36^{\frac{1}{6}} \times \left(\frac{13^6}{28}\right)^{\frac{1}{6}}\right) \]
\[ GM' = \left(36^{\frac{1}{6}}\right) \times \left(13\right) \times \left(\frac{1}{28^{\frac{1}{6}}}\right) \]

8. **Calculate the values**:
- \( 36^{\frac{1}{6}} = 6^{\frac{1}{3}} \)
- \( 28^{\frac{1}{6}} = 14^{\frac{1}{3}} \)

Therefore, we can express the new GM as:
\[ GM' = 13 \times \frac{6^{\frac{1}{3}}}{14^{\frac{1}{3}}} \]
\[ GM' = 13 \times \left(\frac{6}{14}\right)^{\frac{1}{3}} = 13 \times \left(\frac{3}{7}\right)^{\frac{1}{3}} \]

9. **Final result**:
Thus, the correct geometric mean is:
\[ GM' = 13 \times \left(\frac{9}{7}\right)^{\frac{1}{6}} \]

### Final Answer:
The correct geometric mean is \( 13 \times \frac{9}{7} \).

---