This image shows a normal distribution of pregnancy lengths in a population of new mothers, measured in days. The distribution has:

• Mean ($\mu$) = 267 days
• Standard deviation ($\sigma$) = 10 days

The shaded region represents the probability that the pregnancy length falls between 285 and 294 days ($285 < x < 294$).

To find this probability, we calculate the z-scores for the bounds (285 and 294) and use the standard normal distribution.

Step-by-Step Solution:

1. Convert the bounds to z-scores:

   The z-score formula is: $z = \frac{x - \mu}{\sigma}$

   • For $x = 285$: $z_{285} = \frac{285 - 267}{10} = \frac{18}{10} = 1.8$

   • For $x = 294$: $z_{294} = \frac{294 - 267}{10} = \frac{27}{10} = 2.7$

2. Find the probability using the z-scores:

   Using a standard normal distribution table (or a calculator):

   • The cumulative probability for $z = 1.8$ is approximately 0.9641.
   • The cumulative probability for $z = 2.7$ is approximately 0.9965.

3. Calculate the probability of the interval:

   The probability of pregnancy length between 285 and 294 days is: $P(285 < x < 294) = P(z_{294}) - P(z_{285}) = 0.9965 - 0.9641 = 0.0324$

Answer:

The probability that the pregnancy length is between 285 and 294 days is approximately 0.0324, or 3.24%.

Would you like a deeper breakdown or any additional questions answered?

Further Questions

1. How would the probability change if the interval was extended to include values closer to the mean?
2. What does it mean for a value to be within one standard deviation of the mean in a normal distribution?
3. How would the calculation differ if the standard deviation was increased or decreased?
4. What is the probability of a pregnancy lasting less than 267 days?
5. How can we calculate the probability of a pregnancy lasting more than 294 days?

Tip:

In a normal distribution, approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. This is known as the empirical rule.