Problem 16: What proportion of pregnancies lasts between 240 and 280 days? Round answer to four decimal places.

To determine the proportion of pregnancies that last between 240 and 280 days, we typically assume that the duration of pregnancies follows a normal distribution. The average length of a full-term pregnancy is about 280 days, with a standard deviation of approximately 14 days. 1. **Identify the parameters**: - Mean (\(\mu\)) = 280 days - Standard deviation (\(\sigma\)) = 14 days 2. **Convert the days into z-scores**: - For 240 days: \[ z = \frac{X - \mu}{\sigma} = \frac{240 - 280}{14} = \frac{-40}{14} \approx -2.8571 \] - For 280 days: \[ z = \frac{X - \mu}{\sigma} = \frac{280 - 280}{14} = 0 \] 3. **Use the z-table to find the proportions**: - Look up the z-score of -2.8571 in the z-table. This value is approximately 0.0021 (or 0.21%). - The z-score of 0 corresponds to a cumulative probability of 0.5000 (or 50%). 4. **Calculate the proportion of pregnancies between 240 and 280 days**: \[ P(240 < X < 280) = P(Z < 0) - P(Z < -2.8571) = 0.5000 - 0.0021 = 0.4979 \] 5. **Round to four decimal places**: \[ \text{Proportion} \approx 0.4979 \] Thus, the proportion of pregnancies that last between 240 and 280 days is approximately **0.4979**.