First slide

Introduction to statistics

Question

If G is the GM of the product of r sets of observations with geometric means $G_{1}, G_{2}, \dots, G_{r}$ respectively, then G is equal to

Moderate

A

$\log G_{1} + \log G_{2} + \dots + \log G_{r}$
B

$G_{1} \cdot G_{2} \cdot \dots \cdot G_{r}$
C

$\log G_{1} \cdot \log G_{2} \dots \log G_{r}$
D

none of these

Solution

Taking X as the product of varieties $X_{1}, X_{2}, \dots, X_{r}$ corresponding tor sets of observations i. e. $X = X_{1} X_{2} \dots X_{r}$ we have
$\begin{array}{l} \log X = \log X_{1} + \log X_{2} + \dots \log X_{r} \\ \Rightarrow \sum \log X = \sum \log X_{1} + \sum \log X_{2} + \dots + \sum \log X_{r} \\ \Rightarrow \frac{1}{n} \sum \log X = \frac{1}{n} \sum \log X_{1} + \frac{1}{n} \sum \log X_{2} + \dots + \frac{1}{n} \sum \log X_{r} \\ \Rightarrow \log G = \log G_{1} + \log G_{2} + \dots + \log G_{r} \\ \Rightarrow G = G_{1} G_{2} \dots G_{r} . \end{array}$

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