To determine the required sample size for a poll based on a 99% confidence interval with a given margin of error, follow these steps:
1. Identify the key values:
- Margin of error (E): 1.95%
- Confidence level: 99%
- Standard z-score (z\): For a 99% confidence interval, the z-score is 2.576
- Proportion (p): We assume p = 0.5 for maximum variability
- Complement of the proportion (q): [tex]\( q = 1 - p = 0.5 \)[/tex]
2. Use the formula for sample size calculation:
The formula to calculate sample size (n) for a proportion is:
[tex]\[ n = \left( \frac{z^ \times \sqrt{p \times q}}{E} \right)^2 \][/tex]
However, we need to convert the margin of error percentage into a decimal form by dividing by 100:
[tex]\[ E = \frac{1.95}{100} = 0.0195 \][/tex]
3. Substitute the known values into the formula:
[tex]\[ n = \left( \frac{2.576 \times \sqrt{0.5 \times 0.5}}{0.0195} \right)^2 \][/tex]
4. Simplify the expression under the square root:
[tex]\[ \sqrt{0.5 \times 0.5} = \sqrt{0.25} = 0.5 \][/tex]
5. Further simplify the fraction inside the brackets:
[tex]\[ \left( \frac{2.576 \times 0.5}{0.0195} \right)^2 \][/tex]
[tex]\[ = \left( \frac{1.288}{0.0195} \right)^2 \][/tex]
[tex]\[ = (66.05128205128205)^2 \][/tex]
6. Square the result:
[tex]\[ n = (66.05128205128205)^2 \approx 4362.771860618015 \][/tex]
7. Round to the nearest whole number:
[tex]\[ n \approx 4363 \][/tex]
Thus, to achieve a 99% confidence level with a margin of error of 1.95%, approximately 4363 voters should be sampled.
