The lengths of pregnancies in a small rural village are normally distributed with a mean of 269 days and a standard deviation of 16 days.

What is the probability that pregnancy will last fewer than 311 days?
Round your answer to at least three decimals.

Step 1 ：The lengths of pregnancies in a small rural village are normally distributed with a mean of 269 days and a standard deviation of 16 days.

Step 2 ：We are asked to find the probability that a pregnancy will last fewer than 311 days.

Step 3 ：To solve this, we first calculate the z-score, which is given by the formula $z=\frac{X-\mu }{\sigma }$, where $X$ is the value we are interested in, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step 4 ：Substituting the given values, we get $z=\frac{311-269}{16}=2.625$.

Step 5 ：We then use the cumulative distribution function to calculate the probability corresponding to this z-score.

Step 6 ：The probability that a pregnancy will last fewer than 311 days is approximately 0.996. This means that in this small rural village, about 99.6% of pregnancies are expected to last fewer than 311 days.

Step 7 ：Final Answer: $\boxed{{\displaystyle 0.996}}$