If G is the GM of the product of r sets of observations with geometric means `G_(1), G_(2), …,G_(r)` respectively, then G is equal to

Options

A. `log G_(1)+log G_(2) + … + log G_(r)`
B. `G_(1)*G_(2)* …*G_(r)`
C. `log G_(1)*log G_(2) … log G_(r)`
D. none of these

Answer

To find the geometric mean \( G \) of the product of \( r \) sets of observations with geometric means \( G_1, G_2, \ldots, G_r \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Product of Observations**:
Let the product of the numbers in the \( r \) sets of observations be represented as:
\[ X = X_1 \times X_2 \times X_3 \times \ldots \times X_r \]

2. **Take the Logarithm**:
To simplify the multiplication, we take the logarithm of both sides:
\[ \log X = \log(X_1 \times X_2 \times X_3 \times \ldots \times X_r) \]

3. **Apply Logarithmic Properties**:
Using the property of logarithms that states \( \log(a \times b) = \log a + \log b \), we can expand the right-hand side:
\[ \log X = \log X_1 + \log X_2 + \log X_3 + \ldots + \log X_r \]

4. **Divide by \( n \)**:
If we have \( n \) observations in each set, we divide the entire equation by \( n \):
\[ \frac{\log X}{n} = \frac{\log X_1}{n} + \frac{\log X_2}{n} + \frac{\log X_3}{n} + \ldots + \frac{\log X_r}{n} \]

5. **Relate to Geometric Mean**:
The geometric mean \( G \) of the observations can be defined as:
\[ \log G = \frac{\log X}{n} \]
Thus, we can rewrite the equation as:
\[ \log G = \frac{\log X_1}{n} + \frac{\log X_2}{n} + \frac{\log X_3}{n} + \ldots + \frac{\log X_r}{n} \]

6. **Express in Terms of Geometric Means**:
Since \( G_1, G_2, \ldots, G_r \) are the geometric means of the respective sets, we have:
\[ \log G = \log G_1 + \log G_2 + \ldots + \log G_r \]

7. **Combine the Logarithms**:
Using the property \( \log A + \log B = \log(AB) \), we can combine the logarithms:
\[ \log G = \log(G_1 \times G_2 \times \ldots \times G_r) \]

8. **Exponentiate to Solve for \( G \)**:
By exponentiating both sides, we eliminate the logarithm:
\[ G = G_1 \times G_2 \times \ldots \times G_r \]

### Final Result:
Thus, the geometric mean \( G \) of the product of \( r \) sets of observations is given by:
\[ G = G_1 \times G_2 \times \ldots \times G_r \]