Question
A survey was taken of 850 people in a neighborhood, and the results were that 230 people were planning on voting for a particular candidate. Determine the margin of error for a 99% confidence interval of population proportion of people in a neighborhood who were planning on voting for this candidate. Round it to the nearest hundredth and interpret its meaning in context.(2 points) The margin of error=. With 99% confidence, the population proportion of people in a neighborhood who are planning on voting for a particular candidate falls between and .
Answers
GPT-4o mini
To determine the margin of error for a 99% confidence interval of the population proportion of people planning to vote for a particular candidate, we can follow these steps:
1. **Calculate the sample proportion \((\hat{p})\)**:
\[
\hat{p} = \frac{x}{n} = \frac{230}{850} \approx 0.2706
\]
2. **Calculate the standard error (SE) of the sample proportion**:
The standard error of the sample proportion is given by:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.2706(1 - 0.2706)}{850}} \approx \sqrt{\frac{0.2706 \times 0.7294}{850}} \approx \sqrt{\frac{0.1976}{850}} \approx \sqrt{0.000232} \approx 0.0152
\]
3. **Find the critical value**:
For a 99% confidence level, the critical value \(z^*\) can be found using a z-table or standard normal distribution. The critical value for a 99% confidence interval is approximately:
\[
z^* \approx 2.576
\]
4. **Calculate the margin of error (ME)**:
The margin of error is calculated as:
\[
ME = z^* \times SE \approx 2.576 \times 0.0152 \approx 0.0392
\]
5. **Round the margin of error**:
Rounding 0.0392 to the nearest hundredth gives:
\[
ME \approx 0.04
\]
6. **Calculate the confidence interval**:
The lower and upper bounds of the confidence interval for the population proportion can be calculated as:
\[
\hat{p} - ME \quad \text{and} \quad \hat{p} + ME
\]
So,
\[
0.2706 - 0.0392 \approx 0.2314 \quad \text{and} \quad 0.2706 + 0.0392 \approx 0.3098
\]
Thus, we can conclude:
- **Margin of error = 0.04**.
- With 99% confidence, the population proportion of people in the neighborhood who are planning on voting for a particular candidate falls between **0.2314 (or 23.14%)** and **0.3098 (or 30.98%)**.
### Interpretation:
This means that we are 99% confident that the true proportion of all residents in the neighborhood who plan to vote for the candidate lies between approximately 23.14% and 30.98%.
1. **Calculate the sample proportion \((\hat{p})\)**:
\[
\hat{p} = \frac{x}{n} = \frac{230}{850} \approx 0.2706
\]
2. **Calculate the standard error (SE) of the sample proportion**:
The standard error of the sample proportion is given by:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.2706(1 - 0.2706)}{850}} \approx \sqrt{\frac{0.2706 \times 0.7294}{850}} \approx \sqrt{\frac{0.1976}{850}} \approx \sqrt{0.000232} \approx 0.0152
\]
3. **Find the critical value**:
For a 99% confidence level, the critical value \(z^*\) can be found using a z-table or standard normal distribution. The critical value for a 99% confidence interval is approximately:
\[
z^* \approx 2.576
\]
4. **Calculate the margin of error (ME)**:
The margin of error is calculated as:
\[
ME = z^* \times SE \approx 2.576 \times 0.0152 \approx 0.0392
\]
5. **Round the margin of error**:
Rounding 0.0392 to the nearest hundredth gives:
\[
ME \approx 0.04
\]
6. **Calculate the confidence interval**:
The lower and upper bounds of the confidence interval for the population proportion can be calculated as:
\[
\hat{p} - ME \quad \text{and} \quad \hat{p} + ME
\]
So,
\[
0.2706 - 0.0392 \approx 0.2314 \quad \text{and} \quad 0.2706 + 0.0392 \approx 0.3098
\]
Thus, we can conclude:
- **Margin of error = 0.04**.
- With 99% confidence, the population proportion of people in the neighborhood who are planning on voting for a particular candidate falls between **0.2314 (or 23.14%)** and **0.3098 (or 30.98%)**.
### Interpretation:
This means that we are 99% confident that the true proportion of all residents in the neighborhood who plan to vote for the candidate lies between approximately 23.14% and 30.98%.