Final answer:
To be in the top 15% of all human pregnancy durations, the pregnancy would need to last approximately 283 days. This calculation is based on the z-score for the top 85% of a standard normal distribution and the given mean and standard deviation of pregnancy durations.
Step-by-step explanation:
The question asks to determine the number of days a human pregnancy must last to be in the top 15% of all pregnancy durations. Given that the duration of human pregnancies is normally distributed with a mean (μ) of 266 days and a standard deviation (σ) of 16 days, we first need to find the z-score that corresponds to the top 15% of a standard normal distribution. Then, we can use the z-score to calculate the corresponding duration in days.
To find the z-score that cuts off the top 15% of the standard normal distribution, we can use a z-table or calculator. We look for the value that corresponds to the cumulative area of 0.8500 (since the top 15% is equivalent to 100% - 15% = 85%). This value is approximately z = 1.036.
With the z-score and the information on the mean and standard deviation of the pregnancy duration, we apply the formula:
Duration = μ + z × σ
Substituting the given values, we get:
Duration = 266 + 1.036 × 16
Duration ≈ 266 + 16.576
Duration ≈ 282.576 days
Therefore, to be in the top 15% of all pregnancy durations, the pregnancy would need to be approximately 283 days long.