The positive observations are $160,000, $250,000. Compute their simple geometric mean (product of all observations)^(1/n). Report to the nearest dollar.
Correct Answer
D) $200,000
160000×250000 raised to the power 1/2 = 200000. The units remain dollars; this differs from their arithmetic average.
Why This Is the Correct Answer
Option D is correct because $160,000 × $250,000 = 40,000,000,000, and the square root of that product is $200,000. This follows the definition (product of all observations)^(1/n) with n = 2. It differs from the arithmetic average of the two prices.
Why the Other Options Are Wrong
Option A: $205,000
Option A, $205,000, is the arithmetic mean rather than the geometric mean: ($160,000 + $250,000) ÷ 2 = $205,000. The geometric mean is the square root of $160,000 × $250,000, which is $200,000.
Option B: $410,000
Option B, $410,000, is simply the sum of the two prices, $160,000 + $250,000. The geometric mean multiplies the values and takes the square root, giving $200,000.
Option C: $160,000
Option C, $160,000, is just the lower of the two observations. The geometric mean must fall between the two values; the square root of $160,000 × $250,000 is $200,000.
Multiply, then root
Arithmetic mean adds then divides; geometric mean multiplies then takes the root.
How to use: Use this pairing to keep the two averages apart on calculation questions.
Exam Tip
For positive values that differ, the geometric mean is lower than the arithmetic mean, so an answer equal to the simple average is a distractor.
Common Mistakes to Avoid
- -Computing the arithmetic mean instead of the geometric mean
- -Forgetting to take the root after multiplying
- -Adding the values instead of multiplying them
Concept Deep Dive
Analysis
This question tests the geometric mean of two positive values. Multiply the observations and take the nth root, here the square root because n = 2. The result is in dollars and is smaller than the arithmetic mean whenever the values differ.
Background Knowledge
The geometric mean multiplies n positive values and takes the nth root. It is useful for averaging growth rates and ratios because it reflects multiplicative change. For positive values that differ, the geometric mean is lower than the arithmetic mean.
Real-World Application
A market analyst comparing two price levels for the same house type reports a geometric mean of $200,000. She notes that it sits slightly below the simple average of $205,000, as expected for a multiplicative average.
More Statistics Questions
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