calculate the sample size of the population size (N) 340 then margin of error (e) 0.04

To calculate the required sample size for a given population, margin of error, and assuming a 95% confidence level (which corresponds to a Z-score of 1.96), the formula is:

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

Where:
- \( N \) = Population size = 340
- \( e \) = Margin of error = 0.04
- \( Z \) = Z-value corresponding to the desired confidence level (1.96 for 95% confidence)
- \( p \) = Estimated proportion of the population (typically 0.5 is used if unknown to maximize sample size)

### Step-by-Step Calculation

1. **Calculate the numerator:**
   \[
   Z^2 \cdot p \cdot (1 - p) = 1.96^2 \cdot 0.5 \cdot (1 - 0.5) = 1.96^2 \cdot 0.25 = 3.8416 \cdot 0.25 = 0.9604
   \]

   Then:
   \[
   N \cdot Z^2 \cdot p \cdot (1 - p) = 340 \cdot 0.9604 = 326.536
   \]

2. **Calculate the denominator:**
   \[
   e^2 \cdot (N - 1) + Z^2 \cdot p \cdot (1 - p) = 0.04^2 \cdot 339 + 0.9604
   \]
   \[
   0.0016 \cdot 339 + 0.9604 = 0.5424 + 0.9604 = 1.5028
   \]

3. **Calculate the sample size:**
   \[
   n = \frac{326.536}{1.5028} \approx 217.28
   \]

The sample size required is approximately **217**.

Would you like more details or have any questions about this calculation? Here are some related questions you might find useful:

1. How does changing the margin of error affect the sample size?
2. What happens if we change the confidence level from 95% to 99%?
3. How does the population size impact the sample size?
4. What is the significance of the proportion \( p \) in the sample size calculation?
5. Can we adjust the sample size if we know the exact proportion \( p \)?

**Tip:** When unsure of the population proportion, using 0.5 for \( p \) will give you the largest possible sample size needed.

Discover how to calculate the sample size for a population of 340 with a 4% margin of error. This guide includes a detailed step-by-step solution, using key statistical formulas and concepts like the Central Limit Theorem and normal distribution. Ideal for undergraduate statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation