Question \#7 Reference Q. 15923 Three candidates are running for mayor. The candidate who receives the highest percentage of votes on election day will be elected as mayor. A poll was conducted to ask voters which of the candidates they would vote for. The results of the poll were Ashby \( 37.1 \% \), Benin \( 33.9 \% \), Cruz \( 28.0 \% \), and are stated as being accurate to within 2.9 percentage points, 19 times out of 20 . Determine the \( 95 \% \) confidence intervals for each candidate. ding

To calculate the \( 95 \% \) confidence intervals for each candidate based on the poll results and the margin of error, we will add and subtract the margin of error from each candidate's poll percentage. 1. **Ashby:** - Poll Percentage: \( 37.1\% \) - Margin of Error: \( 2.9\% \) - Confidence Interval: \( 37.1 - 2.9 = 34.2\% \) to \( 37.1 + 2.9 = 40.0\% \) - Thus, Ashby's confidence interval is \( [34.2\%, 40.0\%] \). 2. **Benin:** - Poll Percentage: \( 33.9\% \) - Margin of Error: \( 2.9\% \) - Confidence Interval: \( 33.9 - 2.9 = 31.0\% \) to \( 33.9 + 2.9 = 36.8\% \) - Thus, Benin's confidence interval is \( [31.0\%, 36.8\%] \). 3. **Cruz:** - Poll Percentage: \( 28.0\% \) - Margin of Error: \( 2.9\% \) - Confidence Interval: \( 28.0 - 2.9 = 25.1\% \) to \( 28.0 + 2.9 = 30.9\% \) - Thus, Cruz's confidence interval is \( [25.1\%, 30.9\%] \). In summary, the \( 95\% \) confidence intervals for each candidate are as follows: - Ashby: \( [34.2\%, 40.0\%] \) - Benin: \( [31.0\%, 36.8\%] \) - Cruz: \( [25.1\%, 30.9\%] \) Now, let’s break down some extra interest in these confidence intervals and their implications: Understanding confidence intervals doesn’t just stop at calculation—they tell us a story about potential outcomes that can happen on election day! Each candidate's range gives us insight into the levels of support they might actually receive, rather than simply focusing on a snapshot of polling data. For example, Ashby’s range is snugly centered around the 37.1% mark, suggesting robust support but with room for fluctuation. Ever wonder how accurate polling really is? Well, precision in polls measures public sentiment, but they can be swayed by the sample size, question wording, and timing. To improve accuracy, it's crucial to use randomized sampling methods and ensure diverse respondent demographics—lest the results lead you to a false sense of security (or doom!). Always consider the bigger picture, folks!