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To calculate the third quartile (Q3) based on the given data, we follow these steps:

### Step 1: Determine Q3's position
Q3 is the value that separates the top 25% of the data from the bottom 75%. Its position in a cumulative frequency distribution is given by:

\[
\text{Position of Q3} = \frac{3}{4} \times N
\]

Where \(N\) is the total number of observations. In this case:
\[
N = 70 \implies \text{Position of Q3} = \frac{3}{4} \times 70 = 52.5
\]

This means that Q3 lies between the 52nd and 53rd observations in the cumulative distribution.

---

### Step 2: Construct a cumulative frequency table
From the given frequency data:

| Range (cm) | Frequency (f) | Cumulative Frequency |
|------------|---------------|-----------------------|
| 38-41      | 14            | 14                    |
| 42-45      | 20            | 34                    |
| 46-49      | 15            | 49                    |
| 50-53      | 8             | 57                    |
| 54-57      | 5             | 62                    |
| 58-61      | 2             | 64                    |
| 62-65      | 1             | 65                    |

From this table, we observe:
- The 52nd and 53rd observations fall into the range **50-53 cm**, as the cumulative frequency up to 49 is less than 52.5, and the cumulative frequency up to 57 exceeds 52.5.

---

### Step 3: Apply the formula for Q3
To find the exact value of Q3 within the class interval \(50-53\), use the following formula:

\[
Q3 = L + \left(\frac{\frac{3}{4}N - CF}{f}\right) \cdot w
\]

Where:
- \(L = 49.5\), the lower boundary of the class interval \(50-53\),
- \(N = 70\),
- \(CF = 49\), the cumulative frequency of the class before \(50-53\),
- \(f = 8\), the frequency of the class \(50-53\),
- \(w = 3\), the width of the class interval.

Substituting these values:
\[
Q3 = 49.5 + \left(\frac{52.5 - 49}{8}\right) \cdot 3
\]
\[
Q3 = 49.5 + \left(\frac{3.5}{8}\right) \cdot 3
\]
\[
Q3 = 49.5 + 1.3125 = 50.8125
\]

---

### Step 4: Descriptive statement
The third quartile (Q3) is approximately **50.81 cm**, meaning 75% of the students have chest widths less than or equal to **50.81 cm**, and 25% have chest widths greater than this value.

---

Would you like further details or a chart to visualize this calculation? 

Here are five related questions to explore further:
1. How is the first quartile (Q1) calculated using similar data?
2. How can we use this information to create size categories for uniforms?
3. What is the interquartile range (IQR) for this dataset?
4. How can we determine if the data is skewed using quartiles?
5. How would the calculations change if the frequency distribution were grouped differently?

**Tip:** Always construct a cumulative frequency table to simplify locating quartile positions in grouped data.

Uniformes Huitzil será el fabricante de los nuevos uniformes de los alumnos de cierta escuela. Para decidir las tallas, una de las variables en las que se basarán es la anchura de pecho de los estudiantes. Calcula el 3er Cuartil Q3. Escribe también un enunciado descriptivo usando el indicador calculado. Los datos recabados están en la tabla dada.

Learn how to calculate the third quartile (Q3) for grouped data. This problem focuses on the chest widths of students and demonstrates step-by-step how to use the quartile formula, cumulative frequency, and descriptive statistics to solve the problem. Suitable for Grades 9-11.

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