The geometric mean of 6 observations was calculated as 13. It was later observed that one of the observations was recorded as 28 instead of 36. The correct geometric mean is
Step-by-Step Solution
Key Concept: The geometric mean equals the $k$-th root of the product of all observations, so changing one observation scales the product accordingly.
Let $x_1, x_2, \ldots, x_k$ be the observations with $x_1 = 28$. The product of all observations is $28 \cdot x_2 \cdot x_3 \cdots x_k = 13^k$. When we correct the observation from 28 to 36, the new product becomes $36 \cdot x_2 \cdot x_3 \cdots x_k = \frac{13^k}{28} \cdot 36$. The geometric mean of a set is the $k$-th root of the product, so the corrected geometric mean is $13 \cdot \left(\frac{36}{28}\right)^{1/k} = 13 \cdot \left(\frac{9}{7}\right)^{1/k}$. Using the given conditions, we find the corrected observation is $36$.
Correct Answer: 36