Geometric mean of first group of 5 observations is 8 and that of second group of 4 observations is 1282. Then, grouped geometric mean is.

A
64
B
$32 \sqrt{2}$
C
32
D
None of these

Solution:

Suppose that, G₁ is the GM of first 5 observations = 8 and G₂is the GM of last 4 observations = $128 \sqrt{2}$ Also, let G be the grouped mean

$\begin{array}{l} = \sqrt[9]{(G_{1}^{n_{1}} \times G_{2}^{n_{2}})} = \sqrt[9]{(G_{1}^{n_{1}} \times G_{2}^{n_{2}})} = {(G_{1}^{5} \times G_{2}^{4})}^{1 / 9} \\ = {[8^{5} \times (128 \sqrt{2})^{4}]}^{1 / 9} = {[{(2^{3})}^{5} \times {(2^{7 + \frac{1}{2}})}^{4}]}^{1 / 9} \\ = {(2^{15} \times 2^{30})}^{1 / 9} = 2^{\frac{45}{9}} = 2^{5} = 32 \end{array}$

Geometric mean of first group of 5 observations is 8 and that of second group of 4 observations is $128 \sqrt{2}$ . Then, grouped geometric mean is.

Solution:

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