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Let's break down and solve the problem step-by-step:

---

### **1. Calcular las desviaciones estándar de las variables: Peso (X₁), Altura (X₂), Edad (X₃).**

The formula for the standard deviation is:

\[
\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left( X_i - \bar{X} \right)^2}
\]

We will need:
- The **mean** of each variable: \( \bar{X_1}, \bar{X_2}, \bar{X_3} \)
- The **sum of squared deviations** for each.

#### **Data summary:**
\[
\begin{aligned}
X_1: & \quad 64, 71, 53, 67, 55, 58, 77, 57, 56, 51, 76, 68 \\
X_2: & \quad 57, 59, 49, 62, 51, 50, 55, 48, 52, 42, 61, 57 \\
X_3: & \quad 8, 10, 6, 11, 8, 7, 9, 10, 9, 6, 12, 9 \\
\end{aligned}
\]
- **Sample size**: \( n = 12 \)

Let's calculate:

#### **Means (average) of X₁, X₂, and X₃:**
\[
\bar{X_1} = \frac{64 + 71 + 53 + \dots + 68}{12}
\]
\[
\bar{X_2} = \frac{57 + 59 + 49 + \dots + 57}{12}
\]
\[
\bar{X_3} = \frac{8 + 10 + 6 + \dots + 9}{12}
\]

Let’s compute these means.

---

### **2. Aplicar la desviación típica S₁,₂,₃ usando X₁, X₂, X₃.**
The formula provided is:

\[
S_{1,23} = \sqrt{\frac{\sum_{i=1}^{12} \left( X_i - X_{i,\text{est}} \right)^2}{N}}
\]

Where:
- \( X_{i,\text{est}} \) is obtained using the **regression equation**:
  \[
  X_1 = 3.65 + 0.855 X_2 + 1.506 X_3
  \]
  For each data point, you will substitute values from \( X_2 \) and \( X_3 \) into the equation to get the estimated \( X_1 \), and then compute the deviation from the observed \( X_1 \).

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### **3. Error típico de estimación de la población:**

Given the formula:

\[
\hat{S}_{1,23} = \sqrt{\frac{N}{N - 3} S_{1,23}}
\]

---

### **4. Coeficiente de determinación (R²):**

The formula for \( R_{1,23}^2 \) is:

\[
R_{1,23}^2 = 1 - \frac{S_{1,23}^2}{S_1^2}
\]

You’ll need both the variance of the residuals \( S_{1,23}^2 \) and the total variance \( S_1^2 \).

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Would you like me to proceed with detailed calculations, including intermediate steps for each part?

Calculate the standard deviations for all involved variables, then apply regression and estimate errors using the given formulas.

Learn how to calculate standard deviations and apply regression analysis to estimate errors using given data on weight, height, and age. This detailed step-by-step solution covers key concepts in standard deviation, regression, and error estimation.

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