Answer :
To determine the required sample size for a 99% confidence interval given the information from the poll, we need to follow these steps:
1. Identify the values:
- The confidence level is 99%. From the provided critical values table, we see that the critical value (z*) for a 99% confidence level is 2.576.
- The proportion (p) reported in the poll is 39%, or 0.39 as a decimal.
- The margin of error (E) is 2.42 percentage points, which we also convert to a decimal to get 0.0242.
2. Use the sample size formula for proportion:
The formula to calculate the required sample size (n) for estimating a population proportion with a certain margin of error (E) at a given confidence level is:
[tex]\[ n = \left( \frac{z^* \times \sqrt{p \times (1 - p)}}{E} \right)^2 \][/tex]
3. Substitute the values into the formula:
[tex]\[ n = \left( \frac{2.576 \times \sqrt{0.39 \times (1 - 0.39)}}{0.0242} \right)^2 \][/tex]
4. Calculate the intermediate steps:
- Calculate the value inside the square root first:
[tex]\[ p \times (1 - p) = 0.39 \times 0.61 = 0.2379 \][/tex]
- Take the square root of 0.2379:
[tex]\[ \sqrt{0.2379} \approx 0.4877 \][/tex]
- Multiply by the critical value (z*):
[tex]\[ 2.576 \times 0.4877 \approx 1.2559 \][/tex]
- Divide by the margin of error (E):
[tex]\[ \frac{1.2559}{0.0242} \approx 51.91 \][/tex]
- Square the result to find the sample size:
[tex]\[ 51.91^2 \approx 2694.2 \][/tex]
5. Round up to the nearest whole number:
Since sample size must be a whole number, we round up 2694.2 to the next whole number which is 2695. Hence, the required sample size is:
[tex]\[ \boxed{2696} \][/tex]
Therefore, to achieve a 99% confidence interval with a margin of error of 2.42 percentage points, at least 2696 voters should be sampled.
1. Identify the values:
- The confidence level is 99%. From the provided critical values table, we see that the critical value (z*) for a 99% confidence level is 2.576.
- The proportion (p) reported in the poll is 39%, or 0.39 as a decimal.
- The margin of error (E) is 2.42 percentage points, which we also convert to a decimal to get 0.0242.
2. Use the sample size formula for proportion:
The formula to calculate the required sample size (n) for estimating a population proportion with a certain margin of error (E) at a given confidence level is:
[tex]\[ n = \left( \frac{z^* \times \sqrt{p \times (1 - p)}}{E} \right)^2 \][/tex]
3. Substitute the values into the formula:
[tex]\[ n = \left( \frac{2.576 \times \sqrt{0.39 \times (1 - 0.39)}}{0.0242} \right)^2 \][/tex]
4. Calculate the intermediate steps:
- Calculate the value inside the square root first:
[tex]\[ p \times (1 - p) = 0.39 \times 0.61 = 0.2379 \][/tex]
- Take the square root of 0.2379:
[tex]\[ \sqrt{0.2379} \approx 0.4877 \][/tex]
- Multiply by the critical value (z*):
[tex]\[ 2.576 \times 0.4877 \approx 1.2559 \][/tex]
- Divide by the margin of error (E):
[tex]\[ \frac{1.2559}{0.0242} \approx 51.91 \][/tex]
- Square the result to find the sample size:
[tex]\[ 51.91^2 \approx 2694.2 \][/tex]
5. Round up to the nearest whole number:
Since sample size must be a whole number, we round up 2694.2 to the next whole number which is 2695. Hence, the required sample size is:
[tex]\[ \boxed{2696} \][/tex]
Therefore, to achieve a 99% confidence interval with a margin of error of 2.42 percentage points, at least 2696 voters should be sampled.
