A ratio study produces ratios of 0.92, 1 and 1.08. Using the sample standard deviation, what is the z-score of the 1.08 ratio, to two decimals?
Correct Answer
B) 1.00
Mean=1, sample s=0.08; z=(1.08−1)/0.08=1. The sample denominator is n−1 and the standardized score is dimensionless. Common errors: dividing by the range instead of the standard deviation gives 0.50; using the population standard deviation (n) instead of n − 1 gives 1.22; dividing by the variance instead of the standard deviation gives 12.50.
Why This Is the Correct Answer
Option B is correct because the mean of 0.92, 1.00 and 1.08 is 1.00, the squared deviations are 0.0064, 0 and 0.0064, and their sum of 0.0128 divided by n−1 = 2 gives a sample variance of 0.0064. The sample standard deviation is the square root, 0.08. The z-score for 1.08 is (1.08 − 1.00) ÷ 0.08 = 1.00, a dimensionless value.
Why the Other Options Are Wrong
Option A: 0.50
Option A, 0.50, divides the deviation by the range instead of the standard deviation: (1.08 − 1.00) ÷ (1.08 − 0.92) = 0.08 ÷ 0.16 = 0.50. The z-score divides by the sample standard deviation, 0.08, giving 1.00.
Option C: 1.22
Option C, 1.22, comes from using the population standard deviation (dividing by n = 3) instead of n−1: √(0.0128 ÷ 3) ≈ 0.0653, and 0.08 ÷ 0.0653 ≈ 1.22. The stem requires the sample formula, which gives s = √(0.0128 ÷ 2) = 0.08 and z = 1.00.
Option D: 12.50
Option D, 12.50, results from dividing by the variance instead of the standard deviation: 0.08 ÷ 0.0064 = 12.50. The z-score divides by the standard deviation, 0.08, giving 0.08 ÷ 0.08 = 1.00.
Distance over spread
A z-score is distance from the mean divided by the spread (standard deviation), not by the variance. Squared units belong to the variance; the z-score uses the square root.
How to use: Use this when a z-score answer looks far too large, which often signals division by the variance.
Exam Tip
Check the sign first: values above the mean give positive z-scores. Then confirm whether the stem specifies the sample (n−1) or population (n) denominator.
Common Mistakes to Avoid
- -Dividing by n instead of n−1 when the sample formula is required
- -Dividing the deviation by the variance rather than the standard deviation
- -Dividing by the range instead of the standard deviation
Concept Deep Dive
Analysis
This question tests two linked steps: computing a sample standard deviation with the n−1 denominator and then converting one observation into a standardized z-score. First find the mean (1.00), then the squared deviations, divide by n−1 to get the variance, take the square root, and finally divide the observation's distance from the mean by that standard deviation.
Background Knowledge
A z-score expresses how many standard deviations an observation lies above or below the mean, computed as (x − mean) ÷ s. The sample standard deviation divides the sum of squared deviations by n−1 before taking the square root. Because the numerator and denominator share the same units, the z-score has no unit.
Real-World Application
An appraiser reviewing a set of sale-to-list ratios wants to know how unusual one ratio is compared with the group. Converting it to a z-score shows it sits one standard deviation above the mean, which is not by itself an extreme reading.
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