The length of human pregnancies from conception to birth can be approximated a normal distribution with a mean of 265 days and a standard deviation of 15 days. What percentage of all pregnancies will last between 250 and 280 days?
## **Core Concept**
The length of human pregnancies can be approximated by a normal distribution, which is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this case, the mean (μ) is 265 days and the standard deviation (σ) is 15 days.
## **Why the Correct Answer is Right**
To find the percentage of pregnancies that will last between 250 and 280 days, we need to calculate the z-scores for these two values and then use a standard normal distribution (Z-table) to find the probabilities. The z-score formula is (z = frac{X - mu}{sigma}), where (X) is the value, (mu) is the mean, and (sigma) is the standard deviation.
For (X = 250): (z = frac{250 - 265}{15} = frac{-15}{15} = -1).
For (X = 280): (z = frac{280 - 265}{15} = frac{15}{15} = 1).
Using a Z-table, the area to the left of (z = 1) is approximately 0.8413, and the area to the left of (z = -1) is approximately 0.1587. The area between (z = -1) and (z = 1) (which corresponds to the percentage of pregnancies lasting between 250 and 280 days) is (0.8413 - 0.1587 = 0.6826) or (68.26%).
## **Why Each Wrong Option is Incorrect**
- **Option A:** This option is incorrect because it suggests a calculation or percentage that does not match the correct calculation of approximately 68.26%.
- **Option B:** This option is incorrect for the same reason as Option A; it does not align with the calculated percentage.
- **Option D:** This option is incorrect as it also does not match the calculated percentage of approximately 68.26%.
## **Clinical Pearl / High-Yield Fact**
A key point to remember is that in a normal distribution, about 68% of the data falls within one standard deviation of the mean. This is often referred to as the 68-95-99.7 rule, where about 68% of the data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean. This rule can be very helpful in quickly estimating probabilities in a normal distribution.
## **Correct Answer: C. 68.26%**