Answer
- **a.** The normal distribution curve shows key ages at z-scores of -2 (20 years), -1 (24 years), 0 (28 years), +1 (32 years), and +2 (36 years).
- **b.** Approximately 476 applicants are above 32 years old.
- **c.** About 2,048 applicants are between 24 and 32 years old.
Solution
Sure, let's tackle each part of the problem step by step.
### Given:
- **Total applicants**: 3,000
- **Mean age (\( \mu \))**: 28 years
- **Standard deviation (\( \sigma \))**: 4 years
---
### **a. Drawing the Normal Distribution Curve**
While I can't provide a visual graph here, I can describe how to sketch the normal distribution curve with the relevant z-scores and raw scores:
1. **Draw the Bell Curve**: Start with the classic symmetric bell-shaped curve representing the normal distribution.
2. **Mark the Mean**:
- **Center Point**: Plot a vertical line at the mean age, \( \mu = 28 \) years.
3. **Identify Key Points Using Z-Scores**:
- **Z-score Formula**: \( z = \frac{X - \mu}{\sigma} \)
- **Common Z-scores**:
- \( z = -2 \) corresponds to \( X = \mu - 2\sigma = 28 - 8 = 20 \) years
- \( z = -1 \) corresponds to \( X = 24 \) years
- \( z = 0 \) (the mean) corresponds to \( X = 28 \) years
- \( z = +1 \) corresponds to \( X = 32 \) years
- \( z = +2 \) corresponds to \( X = 36 \) years
4. **Label the Curve**:
- Place vertical lines at 20, 24, 28, 32, and 36 years.
- Label each line with its corresponding z-score (-2, -1, 0, +1, +2).
**Summary of Key Points on the Curve:**
| Z-score | Age (X) |
|---------|---------|
| -2 | 20 |
| -1 | 24 |
| 0 | 28 |
| +1 | 32 |
| +2 | 36 |
---
### **b. Number of Applicants Above 32 Years Old**
1. **Calculate the Z-score for 32 years**:
\[
z = \frac{32 - 28}{4} = \frac{4}{4} = 1
\]
2. **Find the Probability (P) for \( Z > 1 \)**:
- Using the standard normal distribution table, \( P(Z > 1) \approx 0.1587 \) (15.87%).
3. **Calculate the Number of Applicants**:
\[
\text{Number of applicants} = P(Z > 1) \times \text{Total applicants} = 0.1587 \times 3000 \approx 476 \text{ applicants}
\]
**Answer:** Approximately **476 applicants** are above 32 years old.
---
### **c. Number of Applicants Between 24 and 32 Years Old**
1. **Calculate the Z-scores for 24 and 32 years**:
- For 24 years:
\[
z = \frac{24 - 28}{4} = \frac{-4}{4} = -1
\]
- For 32 years:
\[
z = \frac{32 - 28}{4} = \frac{4}{4} = 1
\]
2. **Find the Probability (P) for \( -1 < Z < 1 \)**:
- Using the standard normal distribution table, \( P(-1 < Z < 1) \approx 0.6826 \) (68.26%).
3. **Calculate the Number of Applicants**:
\[
\text{Number of applicants} = P(-1 < Z < 1) \times \text{Total applicants} = 0.6826 \times 3000 \approx 2,048 \text{ applicants}
\]
**Answer:** Approximately **2,048 applicants** have ages between 24 and 32 years.
---
### **Summary of Answers**
- **a.** Described the normal distribution curve with key z-scores and raw scores.
- **b.** ~476 applicants are above 32 years old.
- **c.** ~2,048 applicants have ages between 24 and 32 years.
Reviewed and approved by the UpStudy tutoring team
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