Explanation:
The relationship between arithmetic, harmonic, and geometric means is given by: arithmetic mean × harmonic mean = (geometric mean)²
Given:
We can solve for harmonic mean: 8.90 × harmonic mean = (8.50)² 8.90 × harmonic mean = 72.25 harmonic mean = 72.25 ÷ 8.90 harmonic mean = 8.11798 ≈ 8.12
Alternatively, we can use the inequality relationship: harmonic mean < geometric mean < arithmetic mean (for positive values that are not all equal). Since geometric mean = 8.50 and arithmetic mean = 8.90, the harmonic mean must be less than 8.50, which eliminates options B (8.63) and C (9.30). Only option A (8.12) satisfies this inequality.
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An analyst evaluates a dataset with eight values. From the dataset, she calculates the geometric mean to be 8.50. If the arithmetic mean is equal to 8.90, the harmonic mean is closest to:
A
8.12.
B
8.63.
C
9.30.
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