Research wants to take a sample size from a population of 4850 with 5% margins of error a. What is the sample size (n)? b. What sampling method should the researcher use in selecting the samples?

### a. Calculating the Sample Size (n)

To determine the sample size (n) required for a population of 4850 with a 5% margin of error, you can use the following formula for sample size in a finite population:

\[
n = \frac{N \cdot Z^2 \cdot p \cdot (1-p)}{E^2 \cdot (N - 1) + Z^2 \cdot p \cdot (1-p)}
\]

Where:
- \(N\) is the population size (4850).
- \(Z\) is the Z-score, corresponding to the confidence level (commonly 1.96 for 95% confidence).
- \(p\) is the estimated proportion of the population (commonly 0.5 if not known).
- \(E\) is the margin of error (0.05).

Let’s calculate:

\[
n = \frac{4850 \cdot 1.96^2 \cdot 0.5 \cdot (1 - 0.5)}{0.05^2 \cdot (4850 - 1) + 1.96^2 \cdot 0.5 \cdot (1 - 0.5)}
\]

Simplifying the expression:

\[
n = \frac{4850 \cdot 3.8416 \cdot 0.25}{0.0025 \cdot 4849 + 0.9604}
\]

\[
n = \frac{4660.004}{12.1225}
\]

\[
n \approx 384
\]

So, the sample size \(n\) is approximately **384**.

### b. Recommended Sampling Method

Given that the population is finite and a specific sample size is required, **Simple Random Sampling (SRS)** is recommended. This method ensures that every individual in the population has an equal chance of being selected, minimizing bias and ensuring the sample is representative of the population.

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Would you like more details on any part of the calculation or explanation?

### Related Questions
1. What is the impact of increasing the confidence level on the sample size?
2. How does changing the margin of error affect the sample size calculation?
3. What are the assumptions made when using the formula for sample size calculation?
4. Can you explain the concept of confidence intervals and how they relate to sample size?
5. What other sampling methods could be considered if Simple Random Sampling is not feasible?
6. How would the sample size change if the population was much larger, e.g., 50,000?
7. What are the limitations of using a 50% proportion in the sample size formula?
8. How does population variability affect the required sample size?

### Tip
When determining sample size, always consider the potential for non-response or attrition and plan accordingly by slightly increasing the calculated sample size.

Learn how to calculate the sample size needed from a population of 4850 with a 5% margin of error. Includes step-by-step calculation using the sample size formula for finite populations and recommendations on sampling methods. Suitable for advanced level studies in statistics.

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