In a Job fair, 3000 applicants applied for a lob. Their mean age was Rund to be 28 with astandard deviation of 4 years.

a. How many applicants are below 20 years old?

b. Low many applicants are below years (32) old? c. How many have ages between 24 and 32 years.?

We can solve these problems using the **normal distribution** and **z-scores**.

The information provided includes:
- Total applicants = 3000
- Mean age (\(\mu\)) = 28 years
- Standard deviation (\(\sigma\)) = 4 years

### Formula for z-score:
\[
z = \frac{x - \mu}{\sigma}
\]
Where:
- \(x\) = the age we are interested in,
- \(\mu\) = mean,
- \(\sigma\) = standard deviation.

Once we calculate the z-scores, we can use the standard normal distribution table (or calculator) to find the proportion of people below a certain age.

---

### a) How many applicants are below 20 years old?

First, calculate the z-score for 20 years:
\[
z = \frac{20 - 28}{4} = \frac{-8}{4} = -2
\]

Using a standard normal distribution table, the proportion of people below a \(z = -2\) is approximately **0.0228**.

Thus, the number of applicants below 20 years:
\[
0.0228 \times 3000 = 68.4 \approx 68 \text{ applicants}.
\]

---

### b) How many applicants are below 32 years old?

Now, calculate the z-score for 32 years:
\[
z = \frac{32 - 28}{4} = \frac{4}{4} = 1
\]

From the z-table, the proportion of people below \(z = 1\) is approximately **0.8413**.

Thus, the number of applicants below 32 years:
\[
0.8413 \times 3000 = 2523.9 \approx 2524 \text{ applicants}.
\]

---

### c) How many applicants are between 24 and 32 years old?

First, calculate the z-score for 24 years:
\[
z = \frac{24 - 28}{4} = \frac{-4}{4} = -1
\]
For \(z = -1\), the proportion is **0.1587**.

Next, calculate the proportion for 32 years, which we already found is **0.8413**.

Now, the proportion of applicants between 24 and 32 years is:
\[
0.8413 - 0.1587 = 0.6826
\]

Thus, the number of applicants between 24 and 32 years:
\[
0.6826 \times 3000 = 2047.8 \approx 2048 \text{ applicants}.
\]

---

### Summary of Results:
a) Below 20 years old: **68 applicants**  
b) Below 32 years old: **2524 applicants**  
c) Between 24 and 32 years old: **2048 applicants**

---

Would you like further details on the calculations or any clarifications?

Here are 5 related questions to explore:

1. What is the probability of an applicant being exactly 28 years old?
2. What if the standard deviation was 5 years instead of 4 years? How would the answers change?
3. How many applicants are above 36 years old?
4. What proportion of applicants are between 20 and 36 years?
5. Can we calculate the number of applicants over 40 years using the normal distribution?

**Tip:** Understanding the z-score is crucial, as it tells us how far a value is from the mean in terms of standard deviations.

In a Job fair, 3000 applicants applied for a job. Their mean age was rounded to be 28 with a standard deviation of 4 years. a. How many applicants are below 20 years old? b. How many applicants are below 32 years old? c. How many have ages between 24 and 32 years?

Solve a normal distribution problem where 3000 job applicants have a mean age of 28 years with a standard deviation of 4 years. Learn how to calculate how many applicants are below 20 years old, below 32 years old, and between 24 and 32 years using z-scores. Ideal for Grades 11-12 or introductory college students.

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