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Chapter 7: Problem 57

A random sample of likely voters showed that \(55 \%\) planned to vote for Candidate \(\mathrm{X}\), with a margin of error of 2 percentage points and with a \(95 \%\) confidence level. a. Use a carefully worded sentence to report the \(95 \%\) confidence interval for the percentage of voters who plan to vote for Candidate \(\mathrm{X}\). b. Is there evidence that Candidate \(\mathrm{X}\) could lose. c. Suppose the survey was taken on the streets of New York City and the candidate was running for U.S. president. Explain how that would affect your conclusion.

Short Answer

Expert verified
a. The \(95 \% \) confidence interval for the percentage of voters who plan to vote for Candidate X is between \(53 \%\) and \(57 \% \).\nb. It's less likely that Candidate X could lose, as their expected support is above \(50 \% \), but the possibility cannot be ruled out.\nc. If the survey was only conducted in New York City, it might not accurately reflect the sentiment of the entire U.S. electorate as it is a geographically and demographically limited sample.

Step by step solution

01

Calculate the Confidence Interval

Based on the given exercise, a \(95 \%\) confidence level with a margin of error of \(2 \%\) is stated. This means the confidence interval can be calculated by subtracting and adding the margin of error from/to the reported percentage. So, the \(95 \%\) confidence interval will be between \(55 \% -2 \% = 53 \%\) and \(55 \% + 2 \% = 57 \% \).
02

Interpret Confidence Interval

The resulting confidence interval means we can say with \(95 \%\) confidence that between \(53 \%\) and \(57 \%\) of all likely voters intend to vote for candidate X.
03

Evaluate Candidate Losing Probability

Judging from our confidence interval, it is unlikely that candidate X could lose. As neither of the values under the confidence interval is less than \(50 \% \), candidate X has more than \(50 \% \) support from all likely voters. However, we also need to keep in mind that these are just estimations and the actual result might differ.
04

Influence of Survey Setup

The fact that the survey was conducted in New York City could potentially bias the results. If candidate X has particularly strong or weaker support in this city than in the rest of the country, the results will not be representative of the entire U.S. electorate. For a truly representative indicator, a broader geographic sampling would be needed.

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