Question
LESSON 5 Back to Intro Mark as Complete Conclusions from Sample Data Margin of Error Estimating Population Proportion Page I a Item 1 COURSE A survey was taken of 850 people in a neighborhood, and the Dage 19 of 20 OUTLINE results were that 230 people were planning on voting for a Er particular candidate. Determine the margin of error for a 99% Item 2 confidence interval of population proportion of people in a neighborhood who were planning on voting for this candidate. Item 3 COURSE TOOLS Round it to the nearest hundredth and interpret its meaning in context. (2 points) Item 4 Item 5 LESSON < The margin of error =□. FEEDBACK . 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 calculate the margin of error for a 99% confidence interval of the population proportion, we can use the following steps:
1. **Determine the sample proportion (p̂)**:
- The number of people planning to vote for the candidate is 230 out of 850.
- \( p̂ = \frac{230}{850} \approx 0.2706 \)
2. **Calculate the standard error (SE)** of the sample proportion:
- The standard error for a proportion is calculated using the formula:
\[
SE = \sqrt{\frac{p̂(1 - p̂)}{n}}
\]
- Where \( n \) is the sample size.
\[
SE = \sqrt{\frac{0.2706(1 - 0.2706)}{850}} \approx \sqrt{\frac{0.2706 \times 0.7294}{850}} \approx \sqrt{\frac{0.1971}{850}} \approx \sqrt{0.0002325} \approx 0.01525
\]
3. **Determine the z-score for 99% confidence**:
- For a 99% confidence interval, the z-score is approximately 2.576.
4. **Calculate the margin of error (ME)**:
- The margin of error is determined using the formula:
\[
ME = z \times SE
\]
\[
ME = 2.576 \times 0.01525 \approx 0.0393
\]
5. **Round the margin of error to the nearest hundredth**:
- \( ME \approx 0.04 \)
6. **Interpret the meaning of the margin of error in context**:
- With 99% confidence, we can say that the proportion of people in the neighborhood who are planning to vote for this particular candidate falls within the range of \( p̂ \pm ME \).
- Calculate the confidence interval:
- Lower limit: \( 0.2706 - 0.04 = 0.2306 \)
- Upper limit: \( 0.2706 + 0.04 = 0.3106 \)
Thus, the margin of error is approximately **0.04**.
**Final Interpretation**:
With 99% confidence, the population proportion of people in the neighborhood who are planning on voting for this particular candidate falls between approximately **23.06%** and **31.06%**.
1. **Determine the sample proportion (p̂)**:
- The number of people planning to vote for the candidate is 230 out of 850.
- \( p̂ = \frac{230}{850} \approx 0.2706 \)
2. **Calculate the standard error (SE)** of the sample proportion:
- The standard error for a proportion is calculated using the formula:
\[
SE = \sqrt{\frac{p̂(1 - p̂)}{n}}
\]
- Where \( n \) is the sample size.
\[
SE = \sqrt{\frac{0.2706(1 - 0.2706)}{850}} \approx \sqrt{\frac{0.2706 \times 0.7294}{850}} \approx \sqrt{\frac{0.1971}{850}} \approx \sqrt{0.0002325} \approx 0.01525
\]
3. **Determine the z-score for 99% confidence**:
- For a 99% confidence interval, the z-score is approximately 2.576.
4. **Calculate the margin of error (ME)**:
- The margin of error is determined using the formula:
\[
ME = z \times SE
\]
\[
ME = 2.576 \times 0.01525 \approx 0.0393
\]
5. **Round the margin of error to the nearest hundredth**:
- \( ME \approx 0.04 \)
6. **Interpret the meaning of the margin of error in context**:
- With 99% confidence, we can say that the proportion of people in the neighborhood who are planning to vote for this particular candidate falls within the range of \( p̂ \pm ME \).
- Calculate the confidence interval:
- Lower limit: \( 0.2706 - 0.04 = 0.2306 \)
- Upper limit: \( 0.2706 + 0.04 = 0.3106 \)
Thus, the margin of error is approximately **0.04**.
**Final Interpretation**:
With 99% confidence, the population proportion of people in the neighborhood who are planning on voting for this particular candidate falls between approximately **23.06%** and **31.06%**.