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Difficulty: HardData Types, Distributions, and Descriptive Statistics

An epidemiologic surveillance system monitors the time to complete resolution of symptoms in a cohort of 20,00020,000 healthcare workers who contracted a novel upper respiratory viral infection. The symptom resolution times are normally distributed, demonstrating a mean of 1818 days and a variance of 44 days 2^2.

Based on the empirical (68-95-99.7) rule, what is the expected number of healthcare workers whose symptoms resolved in either less than 1414 days or more than 2424 days?

Answer: 530 workers

Answer

530
The standard deviation is the square root of the variance (4=2\sqrt{4} = 2). The value 1414 is two standard deviations below the mean (18−2×2=1418 - 2 \times 2 = 14), which corresponds to the lower 2.5%2.5\% tail of the normal distribution according to the empirical rule. The value 2424 is three standard deviations above the mean (18+3×2=2418 + 3 \times 2 = 24), which corresponds to the upper 0.15%0.15\% tail. Summing these mutually exclusive probabilities gives 2.65%2.65\%. Multiplying this by the cohort size of 20,00020,000 yields exactly 530530 individuals.

Step-by-Step Solution

1
Calculate the standard deviation
σ=2\sigma = 2 days
The standard deviation is required to apply the empirical rule, and it is the square root of the variance (44 days 2^2).
2
Determine the standard score (z-score) limits
1414 days corresponds to z=−2z = -2; 2424 days corresponds to z=+3z = +3
This establishes how many standard deviations the boundaries are from the mean of 1818.
3
Calculate tail probabilities
Lower tail probability is 2.5%2.5\%; upper tail probability is 0.15%0.15\%
According to the 68-95-99.7 rule, 95%95\% falls within 22 SDs (leaving 2.5%2.5\% per tail), and 99.7%99.7\% falls within 33 SDs (leaving 0.15%0.15\% per tail).
4
Calculate total probability and expected count
530530 workers
The total probability of 2.65%2.65\% (0.02650.0265) multiplied by the 20,00020,000 cohort size yields 530530.

Key Concept

Applying the empirical (68-95-99.7) rule to a normal distribution using variance and standard deviation.
Estimated Time:1m 30s
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Topics

Data Types, Distributions, and Descriptive Statistics

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