The length of human pregnancies from conception to birth approximates a normal distribution with a
mean of 266 days and a standard deviation of 16 days. What proportion of all pregnancies will last
i) less than 250 days?
ii) between 240 and 270?
iii) more than 275 days?


To solve these problems, we need to utilize the properties of the normal distribution. Specifically, we will calculate **z-scores** and use the standard normal distribution table (or a calculator) to find the corresponding proportions.

Given information:
- Mean (\(\mu\)) = 266 days
- Standard deviation (\(\sigma\)) = 16 days

The **z-score** formula is:

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

Where:
- \(X\) is the value we're interested in,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.

---

### (i) **Proportion of pregnancies lasting less than 250 days**

1. Calculate the z-score for 250 days:
\[
z = \frac{250 - 266}{16} = \frac{-16}{16} = -1
\]

2. Using the standard normal distribution table, the probability corresponding to \(z = -1\) is approximately **0.1587**.

So, the proportion of pregnancies lasting **less than 250 days** is **0.1587**, or about **15.87%**.

---

### (ii) **Proportion of pregnancies lasting between 240 and 270 days**

1. Calculate the z-scores for both 240 days and 270 days:
   - For 240 days:
     \[
     z = \frac{240 - 266}{16} = \frac{-26}{16} = -1.625
     \]
   - For 270 days:
     \[
     z = \frac{270 - 266}{16} = \frac{4}{16} = 0.25
     \]

2. Using the z-table:
   - The probability corresponding to \(z = -1.625\) is approximately **0.0525**.
   - The probability corresponding to \(z = 0.25\) is approximately **0.5987**.

3. The proportion between 240 and 270 days is the difference between these probabilities:
   \[
   0.5987 - 0.0525 = 0.5462
   \]

So, the proportion of pregnancies lasting **between 240 and 270 days** is **0.5462**, or about **54.62%**.

---

### (iii) **Proportion of pregnancies lasting more than 275 days**

1. Calculate the z-score for 275 days:
\[
z = \frac{275 - 266}{16} = \frac{9}{16} = 0.5625
\]

2. Using the standard normal distribution table, the probability corresponding to \(z = 0.5625\) is approximately **0.7133**.

3. The proportion of pregnancies lasting **more than 275 days** is:
\[
1 - 0.7133 = 0.2867
\]

So, the proportion of pregnancies lasting **more than 275 days** is **0.2867**, or about **28.67%**.

---

### Summary:
1. **Less than 250 days:** 15.87%
2. **Between 240 and 270 days:** 54.62%
3. **More than 275 days:** 28.67%

---

Do you want more details or have any questions?

Here are 5 related questions:
1. What is the probability of a pregnancy lasting less than 240 days?
2. How can we calculate the probability of a pregnancy lasting exactly 266 days?
3. What proportion of pregnancies lasts between 260 and 280 days?
4. How would a larger standard deviation affect the distribution?
5. What is the median length of pregnancy given the normal distribution?

**Tip:** Always check whether the area under the curve makes sense relative to your mean and z-scores for probability problems involving the normal distribution.

The length of human pregnancies from conception to birth approximates a normal distribution with a mean of 266 days and a standard deviation of 16 days. What proportion of all pregnancies will last i) less than 250 days? ii) between 240 and 270? iii) more than 275 days?

Explore the probability of human pregnancies lasting less than 250 days, between 240 and 270 days, and more than 275 days using normal distribution. Learn how to calculate Z-scores and interpret standard normal probabilities with detailed solutions. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation