The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days.

﻿﻿﻿In a letter to an advice column, a wife claimed to have given birth 309 days after a brief visit from her husband, who was working in another country. Find the probability of a pregnancy lasting 309 days or longer. What does the result suggest?

﻿﻿﻿If the length of pregnancy is in the lowest 4%, then the baby is premature. Find the length that separates premature babies from those who are not considered premature.

a. The probability that a pregnancy will last 309 days or longer is (Round to four decimal places as needed.)

We are dealing with a normally distributed variable for the lengths of pregnancies, with:

- Mean (\(\mu\)) = 266 days
- Standard deviation (\(\sigma\)) = 15 days

### Part 1: Probability of a Pregnancy Lasting 309 Days or Longer
To find the probability that a pregnancy lasts 309 days or longer, we need to calculate the **Z-score** for 309 days. The Z-score is given by the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \(X = 309\) (the given pregnancy length),
- \(\mu = 266\) (mean),
- \(\sigma = 15\) (standard deviation).

Substitute the values:

\[
Z = \frac{309 - 266}{15} = \frac{43}{15} = 2.87
\]

Now, using the Z-score of 2.87, we can find the corresponding probability from the standard normal distribution table (or use a calculator). The area to the left of \(Z = 2.87\) is approximately 0.9979.

Since we are looking for the probability of the pregnancy lasting **309 days or longer**, we want the area to the **right** of \(Z = 2.87\), which is:

\[
P(X \geq 309) = 1 - 0.9979 = 0.0021
\]

Thus, the probability of a pregnancy lasting 309 days or longer is approximately **0.0021**, or **0.21%**.

### Interpretation
This result suggests that a pregnancy lasting 309 days or longer is highly unusual. In fact, the probability of such an event is only about 0.21%, which implies that the claim of the wife giving birth after 309 days is statistically very unlikely.

---

### Part 2: Length That Separates Premature Babies
To find the length of pregnancy that separates premature babies (lowest 4%), we need to find the pregnancy length corresponding to the **4th percentile** of the normal distribution.

From standard normal distribution tables, the Z-score corresponding to the lowest 4% is approximately **-1.75**.

Now, we use the Z-score formula to find the corresponding length of pregnancy:

\[
Z = \frac{X - \mu}{\sigma}
\]

Substitute the known values:

\[
-1.75 = \frac{X - 266}{15}
\]

Solve for \(X\):

\[
X - 266 = -1.75 \times 15 = -26.25
\]

\[
X = 266 - 26.25 = 239.75
\]

Thus, the pregnancy length that separates premature babies is approximately **239.75 days**.

---

### Summary
a. The probability that a pregnancy lasts 309 days or longer is approximately **0.0021** (or 0.21%).

b. The length of pregnancy that separates premature babies from those who are not is approximately **239.75 days**.

Would you like more details on any of the steps or explanations?

### Related Questions:
1. What are other factors that can influence pregnancy duration aside from biological variation?
2. How would the probability change if the mean and standard deviation were different for a specific population?
3. What is the significance of calculating percentiles in normally distributed data?
4. How would the interpretation change if the probability of a 309-day pregnancy were higher?
5. Can the Z-score approach be used to estimate probabilities in other biological or medical contexts?

### Tip: 
When working with normal distributions, understanding the Z-score is key, as it allows you to standardize any normal variable and use standard normal tables for probability calculations.

The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days. In a letter to an advice column, a wife claimed to have given birth 309 days after a brief visit from her husband, who was working in another country. Find the probability of a pregnancy lasting 309 days or longer. What does the result suggest? If the length of pregnancy is in the lowest 4%, then the baby is premature. Find the length that separates premature babies from those who are not considered premature.

Explore how to calculate the probability of a pregnancy lasting 309 days or longer, and find the threshold for premature babies using normal distribution. This problem covers key concepts such as Z-scores, percentiles, and the normal distribution theorem. Suitable for students in Grades 10-12.

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