𝑛 =
(𝑍1−
𝛼
2
)
2
∙ 𝑝̂∙ 𝑞̂ ∙ 𝑁
𝑒
2 ∙ (𝑁 − 1)+ (𝑍1−
𝛼
2
)
2
∙ 𝑝̂∙ 𝑞̂

genera un ejemplo con esta formula

Let's work through an example using the provided sample size formula:

\[
n = \frac{{(Z_{1-\frac{\alpha}{2}})^2 \cdot \hat{p} \cdot \hat{q} \cdot N}}{{e^2 \cdot (N - 1) + (Z_{1-\frac{\alpha}{2}})^2 \cdot \hat{p} \cdot \hat{q}}}
\]

### Example Scenario:

- **Population Size (N)**: 10,000
- **Desired Confidence Level**: 95% (which gives us \( Z_{1-\frac{\alpha}{2}} = 1.96 \))
- **Estimated Proportion (\(\hat{p}\))**: 0.5 (assuming maximum variability)
- **Complement of Proportion (\(\hat{q} = 1 - \hat{p}\))**: 0.5
- **Margin of Error (e)**: 0.05 (5%)

### Step-by-Step Calculation:

1. **Calculate the numerator**:

   \[
   (Z_{1-\frac{\alpha}{2}})^2 \cdot \hat{p} \cdot \hat{q} \cdot N = (1.96)^2 \cdot 0.5 \cdot 0.5 \cdot 10,000
   \]

   \[
   = 3.8416 \cdot 0.25 \cdot 10,000 = 9604
   \]

2. **Calculate the denominator**:

   \[
   e^2 \cdot (N - 1) + (Z_{1-\frac{\alpha}{2}})^2 \cdot \hat{p} \cdot \hat{q} 
   \]

   \[
   = (0.05)^2 \cdot (10,000 - 1) + 3.8416 \cdot 0.25 
   \]

   \[
   = 0.0025 \cdot 9999 + 0.9604 = 25 + 0.9604 = 25.9604
   \]

3. **Calculate the sample size (n)**:

   \[
   n = \frac{9604}{25.9604} \approx 369.8
   \]

### Result:

The calculated sample size \(n\) is approximately **370**. 

This means that for a population of 10,000 people, with a confidence level of 95%, an estimated proportion of 0.5, and a margin of error of 5%, you would need a sample size of 370 individuals.

---

Would you like further details on any of these steps or have other questions?

---

**Relative Questions**:
1. How does the sample size change if the population size (N) increases?
2. What impact does reducing the margin of error have on the sample size?
3. How do you interpret the Z-score in the context of confidence levels?
4. What if the estimated proportion \(\hat{p}\) is different, say 0.7 instead of 0.5?
5. How would the formula change if the population size was unknown?

**Tip**: Always double-check your confidence level and margin of error as they have significant effects on the required sample size.

𝑛 = (𝑍1− 𝛼/2)² ∙ 𝑝̂∙ 𝑞̂ ∙ 𝑁 / [𝑒² ∙ (𝑁 − 1) + (𝑍1− 𝛼/2)² ∙ 𝑝̂∙ 𝑞̂] genera un ejemplo con esta formula

Learn how to calculate the required sample size using a confidence interval formula for a population of 10,000. This example walks through the steps to determine the sample size with a 95% confidence level, assuming maximum variability. Suitable for university-level statistics students.

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