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 1: Understand the given information
We know that the geometric mean (GM) of six observations was calculated as 13. This means:

\[ GM = \sqrt[6]{a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6} = 13 \]

### Step 2: Calculate the product of the observations
From the geometric mean formula, we can express the product of the observations:

\[ a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6 = 13^6 \]

### Step 3: Identify the incorrect observation
One of the observations was recorded as 28 instead of 36. Let's denote the incorrect observation as \( a_1 = 28 \) and the correct observation as \( a_1 = 36 \).

### Step 4: Express the product with the incorrect observation
Using the incorrect observation, we have:

\[ 28 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6 = 13^6 \]

### Step 5: Calculate the product of the remaining observations
From the above equation, we can express the product of the remaining observations:

\[ a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6 = \frac{13^6}{28} \]

### Step 6: Calculate the product with the correct observation
Now, substituting the correct observation, we have:

\[ 36 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6 = 36 \cdot \frac{13^6}{28} \]

### Step 7: Calculate the new product
Thus, the new product of the observations with the correct value is:

\[ a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6 = \frac{36 \cdot 13^6}{28} \]

### Step 8: Calculate the new geometric mean
Now we can find the new geometric mean:

\[ GM_{correct} = \sqrt[6]{36 \cdot \frac{13^6}{28}} = \sqrt[6]{\frac{36 \cdot 13^6}{28}} \]

### Step 9: Simplify the expression
We can simplify this expression:

\[ GM_{correct} = \sqrt[6]{\frac{36}{28}} \cdot 13 \]

### Step 10: Simplify further
Calculating \( \frac{36}{28} = \frac{9}{7} \):

\[ GM_{correct} = \sqrt[6]{\frac{9}{7}} \cdot 13 \]

### Final Result
Thus, the correct geometric mean is:

\[ GM_{correct} = 13 \cdot \left(\frac{9}{7}\right)^{\frac{1}{6}} \]