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Difficulty: MediumData Types, Distributions, and Descriptive Statistics

A pediatric researcher is analyzing the birth weights of full-term neonates born at 39 weeks of gestation at a large regional hospital. The dataset, which includes 5,000 infants, follows a normal distribution. The analysis reveals a mean birth weight of 3,4003,400 g and a variance of 90,000 g290,000 \text{ g}^2. For a secondary analysis, the researcher needs to identify the birth weight threshold for the lowest 2.5%2.5\% of the cohort. Based on the empirical rule, what is the maximum birth weight (in grams) that would fall into the lowest 2.5%2.5\% of this distribution?

Answer: 2800 g

Answer

2800
The correct threshold is 2,8002,800 g. In a normal distribution, the empirical (68-95-99.7) rule states that approximately 95%95\% of the data falls within two standard deviations (22 SD) of the mean. This leaves 5%5\% of the data in the tails, evenly distributed as 2.5%2.5\% in the lower tail and 2.5%2.5\% in the upper tail. Therefore, the lowest 2.5%2.5\% of neonates corresponds to birth weights that are exactly 22 SD below the mean. The standard deviation is the square root of the variance, so 90,000=300\sqrt{90,000} = 300 g. Subtracting 22 SD from the mean of 3,4003,400 g yields 3,400−2(300)=2,8003,400 - 2(300) = 2,800 g.

Step-by-Step Solution

1
Calculate the standard deviation from the given variance.
90,000=300\sqrt{90,000} = 300 g
The empirical rule and normal distribution properties are based on standard deviation, not variance.
2
Determine the number of standard deviations from the mean corresponding to the 2.5th percentile.
22 standard deviations below the mean.
According to the empirical rule (68-95-99.7), 95%95\% of values lie within 22 standard deviations of the mean. Due to the symmetry of the normal distribution, the remaining 5%5\% is split equally between the two tails, meaning 2.5%2.5\% of values fall below −2-2 standard deviations.
3
Calculate the specific birth weight threshold.
3,400−(2×300)=2,8003,400 - (2 \times 300) = 2,800 g
Subtracting 22 standard deviations from the mean provides the maximum birth weight that falls into the lowest 2.5%2.5\% of the distribution.

Key Concept

Applying the empirical rule (68-95-99.7) to a normal distribution and deriving standard deviation from variance.
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Topics

Data Types, Distributions, and Descriptive Statistics

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