Answer
- **a.** A normal distribution curve centered at 28 years with key points at 24 years (Z = -1), 28 years (Z = 0), and 32 years (Z = +1).
- **b.** Approximately 476 applicants are older than 32 years.
- **c.** Approximately 2,048 applicants are between 24 and 32 years old.
Solution
Let's address each part of the problem step by step.
---
## **Given:**
- **Number of Applicants:** 3,000
- **Mean Age (μ):** 28 years
- **Standard Deviation (σ):** 4 years
---
### **a. Drawing the Normal Curve Distribution with Z-scores and Raw Scores**
While I can't provide a visual diagram here, I can describe how you would sketch the normal distribution curve with the relevant annotations.
1. **Draw the Axes:**
- **Horizontal Axis (X-axis):** Represents the age of applicants.
- **Vertical Axis (Y-axis):** Represents the probability density.
2. **Mark the Mean:**
- Place a vertical line at **μ = 28 years** at the center of the curve.
3. **Identify and Mark Key Raw Scores:**
- **One Standard Deviation Below the Mean (μ - σ):** 28 - 4 = **24 years**
- **One Standard Deviation Above the Mean (μ + σ):** 28 + 4 = **32 years**
- **Z-scores:**
- For 24 years: \( Z = \frac{24 - 28}{4} = -1 \)
- For 28 years: \( Z = \frac{28 - 28}{4} = 0 \)
- For 32 years: \( Z = \frac{32 - 28}{4} = +1 \)
4. **Sketch the Bell Curve:**
- The highest point is at the mean (28 years).
- The curve is symmetric about the mean.
- Mark the points at 24 years (Z = -1) and 32 years (Z = +1).
- You can also extend this to mark ±2σ (20 and 36 years) and ±3σ (16 and 40 years) if desired.
**Visualization Summary:**
```
Age (years)
16 20 24 28 32 36 40
|------|------|------|------|------|------|
-3σ -2σ -1σ μ +1σ +2σ +3σ
```
- **Z-scores** correspond to each σ interval from the mean.
---
### **b. Number of Applicants Above 32 Years Old**
To find the number of applicants older than 32 years:
1. **Calculate the Z-score for 32 years:**
\[
Z = \frac{X - μ}{σ} = \frac{32 - 28}{4} = 1
\]
2. **Find the Probability for Z > 1:**
- Using the standard normal distribution table, \( P(Z > 1) = 0.1587 \) (or 15.87%).
3. **Calculate the Number of Applicants:**
\[
\text{Number of Applicants} = 0.1587 \times 3000 \approx 476.1
\]
- **Approximately 476 applicants** are older than 32 years.
---
### **c. Number of Applicants with Ages Between 24 and 32 Years**
To find the number of applicants aged between 24 and 32 years:
1. **Calculate the Z-scores:**
- For 24 years:
\[
Z = \frac{24 - 28}{4} = -1
\]
- For 32 years:
\[
Z = \frac{32 - 28}{4} = 1
\]
2. **Find the Probability for -1 ≤ Z ≤ 1:**
- From the standard normal distribution table, \( P(-1 ≤ Z ≤ 1) = 0.6826 \) (or 68.26%).
3. **Calculate the Number of Applicants:**
\[
\text{Number of Applicants} = 0.6826 \times 3000 \approx 2047.8
\]
- **Approximately 2,048 applicants** are aged between 24 and 32 years.
---
## **Summary:**
- **a.** A normal distribution curve centered at 28 years with a standard deviation of 4 years. Key points at 24 years (Z = -1), 28 years (Z = 0), and 32 years (Z = +1).
- **b.** **Approximately 476 applicants** are older than 32 years.
- **c.** **Approximately 2,048 applicants** are aged between 24 and 32 years.
Reviewed and approved by the UpStudy tutoring team
Explain
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