A researcher in the Ministry of Education wishes to estimate, with 95% confidence, 
the proportion of teachers willing to work in rural areas. A previous study shows 
that 27.3% of those interviewed were willing to work in rural areas. The researcher 
wishes to be accurate within 5% of the true proportion. Find the minimum sample 
size necessary for this study. 

To find the minimum sample size required for estimating a proportion with a specified level of confidence and margin of error, we can use the formula for sample size in proportion problems:

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

where:
- \( n \) = required sample size
- \( Z \) = Z-score corresponding to the desired confidence level
- \( p \) = estimated proportion from prior studies
- \( E \) = margin of error

### Given values:
- Confidence level = 95% ⟹ \( Z = 1.96 \) (Z-score for 95% confidence level)
- Estimated proportion \( p = 0.273 \) (or 27.3% from previous study)
- Margin of error \( E = 0.05 \) (or 5%)

### Step 1: Substitute the values into the formula
\[
n = \frac{(1.96)^2 \cdot 0.273 \cdot (1 - 0.273)}{(0.05)^2}
\]

### Step 2: Calculate each component
1. Calculate \( Z^2 \):
   \[
   Z^2 = (1.96)^2 = 3.8416
   \]

2. Calculate \( p \cdot (1 - p) \):
   \[
   p \cdot (1 - p) = 0.273 \cdot (1 - 0.273) = 0.273 \cdot 0.727 = 0.198231
   \]

3. Calculate \( E^2 \):
   \[
   E^2 = (0.05)^2 = 0.0025
   \]

### Step 3: Plug these values back into the formula
\[
n = \frac{3.8416 \cdot 0.198231}{0.0025}
\]

\[
n = \frac{0.761275776}{0.0025}
\]

\[
n \approx 304.51
\]

Since the sample size \( n \) must be a whole number, we round up to the next integer.

### Final Answer:
The minimum sample size required is:
\[
n = 305
\]

---

Would you like a more detailed breakdown or have any questions about these calculations?

### Follow-up Questions
1. How would the sample size change if we wanted a smaller margin of error, such as 3%?
2. What would be the impact on the sample size if the confidence level were increased to 99%?
3. Why do we round up when calculating sample size in this context?
4. If no prior study was available, what value of \( p \) should be used to estimate the sample size?
5. How does the sample size formula differ when estimating a mean instead of a proportion?

### Tip
When estimating proportions and unsure of the proportion \( p \), using \( p = 0.5 \) maximizes the required sample size, ensuring a conservative estimate.

A researcher in the Ministry of Education wishes to estimate, with 95% confidence, the proportion of teachers willing to work in rural areas. A previous study shows that 27.3% of those interviewed were willing to work in rural areas. The researcher wishes to be accurate within 5% of the true proportion. Find the minimum sample size necessary for this study.

Discover how to calculate the minimum sample size needed to estimate the proportion of teachers willing to work in rural areas with 95% confidence and a 5% margin of error. Based on a previous study showing 27.3% willingness, the required sample size is 305.

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