Question
You are given a margin of error as 3 percentage points and a confidence level of 99%. If the sample percentage from a recent poll is 35%, find the minimum sample size to estimate a population proportion.
(1 point)
• 496
• 2579
• 1677
• 248
(1 point)
• 496
• 2579
• 1677
• 248
Answers
GPT-4o mini
To calculate the minimum sample size needed to estimate a population proportion with a specified margin of error and confidence level, you can use the following formula:
\[
n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right)
\]
Where:
- \( n \) is the sample size,
- \( Z \) is the Z-score corresponding to the confidence level,
- \( p \) is the sample proportion,
- \( E \) is the margin of error.
### Given Data:
- Margin of error (\( E \)) = 3 percentage points = 0.03
- Confidence level = 99% → Z-score (\( Z \)) ≈ 2.576 (from Z-table)
- Sample proportion (\( p \)) = 35% = 0.35
### Calculation:
1. **Substituting values into the formula**:
\[
n = \frac{(2.576)^2 \cdot 0.35 \cdot (1 - 0.35)}{(0.03)^2}
\]
2. **Calculate \( (2.576)^2 \)**:
\[
(2.576)^2 \approx 6.635776
\]
3. **Calculate \( 0.35 \cdot (1 - 0.35) \)**:
\[
0.35 \cdot 0.65 = 0.2275
\]
4. **Calculate \( (0.03)^2 \)**:
\[
(0.03)^2 = 0.0009
\]
5. **Putting it all together**:
\[
n = \frac{6.635776 \cdot 0.2275}{0.0009}
\]
\[
n \approx \frac{1.50858632}{0.0009} \approx 1676.21
\]
Since the sample size must be a whole number, round up to the nearest whole number:
\[
n \approx 1677
\]
### Conclusion:
The minimum sample size to estimate a population proportion with a margin of error of 3 percentage points at a 99% confidence level is **1677**. Therefore, the correct answer is:
• 1677
\[
n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right)
\]
Where:
- \( n \) is the sample size,
- \( Z \) is the Z-score corresponding to the confidence level,
- \( p \) is the sample proportion,
- \( E \) is the margin of error.
### Given Data:
- Margin of error (\( E \)) = 3 percentage points = 0.03
- Confidence level = 99% → Z-score (\( Z \)) ≈ 2.576 (from Z-table)
- Sample proportion (\( p \)) = 35% = 0.35
### Calculation:
1. **Substituting values into the formula**:
\[
n = \frac{(2.576)^2 \cdot 0.35 \cdot (1 - 0.35)}{(0.03)^2}
\]
2. **Calculate \( (2.576)^2 \)**:
\[
(2.576)^2 \approx 6.635776
\]
3. **Calculate \( 0.35 \cdot (1 - 0.35) \)**:
\[
0.35 \cdot 0.65 = 0.2275
\]
4. **Calculate \( (0.03)^2 \)**:
\[
(0.03)^2 = 0.0009
\]
5. **Putting it all together**:
\[
n = \frac{6.635776 \cdot 0.2275}{0.0009}
\]
\[
n \approx \frac{1.50858632}{0.0009} \approx 1676.21
\]
Since the sample size must be a whole number, round up to the nearest whole number:
\[
n \approx 1677
\]
### Conclusion:
The minimum sample size to estimate a population proportion with a margin of error of 3 percentage points at a 99% confidence level is **1677**. Therefore, the correct answer is:
• 1677