To find the pregnancy length that represents the 85th percentile, we need to use the mean and standard deviation of the normal distribution.
Assuming that the standard deviation of pregnancy lengths is 15 days (a common value used in similar problems), we can standardize the 85th percentile using the Z-score formula:
Z = (X - μ) / σ
where X is the pregnancy length corresponding to the 85th percentile, μ is the mean of the distribution (268 days), and σ is the standard deviation (15 days).
To find the Z-score corresponding to the 85th percentile, we can use a standard normal distribution table or a calculator. For example, using a calculator:
Z = invNorm(0.85) ≈ 1.04
where invNorm() is the inverse normal distribution function.
Now we can use the Z-score formula to solve for X:
1.04 = (X - 268) / 15
Multiplying both sides by 15:
X - 268 = 15 * 1.04
X - 268 = 15.6
Adding 268 to both sides:
X = 268 + 15.6
X ≈ 283.6
Therefore, the pregnancy length that represents the 85th percentile is approximately 283.6 days.