https://r2.erweima.ai/mathbot/image/1723610332150sdcrhot2.jpg

- 𝒆 = 0.03
- 𝜶 = 0.04
- 𝑝̂ = 0.25
- 𝑞̂ = 0.75

To calculate the sample size \( n \) using the provided formula, we'll substitute the given values into the equation:

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

### Given values:
- \( e = 0.03 \)
- \( \alpha = 0.04 \) (This means the confidence level is \( 1 - \alpha = 0.96 \), and thus \( Z_{1-\alpha/2} \) is needed)
- \( \hat{p} = 0.25 \)
- \( \hat{q} = 0.75 \) (since \( \hat{q} = 1 - \hat{p} \))

### Finding \( Z_{1-\alpha/2} \):
From the table provided in the image, \( Z_{1-\alpha/2} \) corresponds to different confidence levels. For \( \alpha = 0.04 \), we have a confidence level of \( 96\% \), so we will look at \( Z_{0.98} \):

\[
Z_{0.98} = 2.05
\]

### Substituting values:
\[
n = \left(\frac{2.05^2 \cdot 0.25 \cdot 0.75}{0.03^2}\right)
\]

Let's calculate this:

\[
n = \left(\frac{4.2025 \cdot 0.25 \cdot 0.75}{0.0009}\right)
\]

\[
n = \left(\frac{4.2025 \cdot 0.1875}{0.0009}\right)
\]

\[
n = \left(\frac{0.78796875}{0.0009}\right)
\]

\[
n = 875.52
\]

### Final Answer:
The sample size \( n \) required is approximately **876** (rounded up to the nearest whole number).

---

Would you like further details or explanations?  
Here are some related questions you might find helpful:

1. What is the significance of the confidence level in determining sample size?
2. How does changing the margin of error \( e \) affect the sample size?
3. What is the relationship between \( \hat{p} \) and \( \hat{q} \) in sample size calculations?
4. How do you interpret the Z-value in a sample size calculation?
5. What happens if the population proportion \( \hat{p} \) is unknown?
6. Can the sample size formula be used for different types of surveys?
7. What is the effect of a higher confidence level on the sample size?
8. How would the sample size change if the margin of error were decreased to 0.02?

**Tip:** Always round up the sample size to the nearest whole number to ensure the sample is large enough to meet the desired confidence level and margin of error.

Learn how to calculate sample size with a confidence level of 96%, margin of error 0.03, and population proportion 0.25 using a detailed step-by-step solution. This problem covers key statistical concepts, including the sample size formula, confidence levels, margin of error, and population proportions. Suitable for advanced college-level students and researchers.

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