A large epidemiological study is investigating a novel serum biomarker, alpha-7-glycoprote… — USMLE Step 3

A large epidemiological study is investigating a novel serum biomarker, alpha-7-glycoprotein (A7G), to predict incident heart failure in adults over age $50$ . The A7G levels in the study population ( $N = 10,000$ ) are determined to follow a normal distribution. Preliminary statistical analysis reveals that $16\%$ of the participants have an A7G level greater than $45 \text{ mg/L}$ , while $2.5\%$ of the participants have an A7G level less than $21 \text{ mg/L}$ .

Based on these findings, what is the variance of the A7G levels in this study population? (Provide your answer as a numeric value)

Cevap: 64 (\text{mg/L})^2

Cevap

The problem requires applying the 68-95-99.7 (empirical) rule for normally distributed data. According to this rule, approximately

68\%

of values lie within

1

standard deviation (SD) of the mean, leaving

32\%

outside this range (

16\%

in each tail). Therefore, the value

45 \text{ mg/L}

, above which

16\%

of the data falls, represents the mean plus

1\text{ SD}

(

\mu + 1\sigma = 45

). Similarly,

95\%

of values lie within

2\text{ SDs}

of the mean, leaving

5\%

outside (

2.5\%

in each tail). The value

21 \text{ mg/L}

, below which

2.5\%

of the data falls, represents the mean minus

2\text{ SDs}

(

\mu - 2\sigma = 21

). Subtracting the second equation from the first yields

3\sigma = 24

, so

\sigma = 8 \text{ mg/L}

. The question asks for the variance, which is the square of the standard deviation (

\sigma^2 = 8^2 = 64

). Thus, the variance is

64

Adım Adım Çözüm

Identify the percentiles given in the clinical vignette and relate them to the empirical rule (68-95-99.7 rule) for a normal distribution.

16% above 45 mg/L corresponds to the top tail outside 1 standard deviation (SD). 2.5% below 21 mg/L corresponds to the bottom tail outside 2 SDs.

In a normal distribution, 68% of data falls within 1 SD of the mean (leaving 32% outside, or 16% in each tail). 95% of data falls within 2 SDs (leaving 5% outside, or 2.5% in each tail).

Translate the percentiles into a system of algebraic equations using the mean (\mu) and standard deviation (\sigma).

Equation 1: \mu + 1\sigma = 45. Equation 2: \mu - 2\sigma = 21.

This establishes the mathematical relationship between the known values and the unknown distribution parameters.

Solve the system of linear equations to find the standard deviation (\sigma).

Subtracting Equation 2 from Equation 1 yields: (\mu + 1\sigma) - (\mu - 2\sigma) = 45 - 21. Therefore, 3\sigma = 24, which means \sigma = 8 mg/L.

Isolating \sigma allows us to determine the standard deviation, which is required to find the variance.

Calculate the variance by squaring the standard deviation.

Variance = \sigma^2 = 8^2 = 64.

Variance is defined mathematically as the square of the standard deviation.

Anahtar Kavram

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

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