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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%. We're 95% confident that the true proportion of voters who plan to vote for Candidate X lies within this range. b. There's no evidence Candidate X would lose from this sample, as all values in the confidence interval are over 50%. c. However, the results might not apply to the entire U.S. population since the survey was done only in New York City, hence Candidate X could potentially lose in other places.

Step by step solution

01

Find the Confidence Interval

The confidence interval, given the margin of error, can be found by simply adding and subtracting the margin of error from the sample percentage. Here, the margin of error is 2% and the sample percentage is 55%. Thus, the confidence interval is from \(55-2 = 53%\) to \(55+2 = 57%\).
02

Interpret the Confidence Interval

The interpretation of this confidence interval is that we're 95% confident that the true proportion of voters who plan to vote for Candidate X lies between 53% and 57%.
03

Decide if Candidate X Could Lose

Based on the confidence interval, since all values are over 50%, there's no evidence Candidate X would lose if we only consider this sample.
04

Consider the Sampling Method

However, the survey was taken in New York City only. If Candidate X is running for a national election like U.S. President, the sample may not be representative of the entire U.S. population. Therefore, the results might not apply to the entire country, and Candidate X could potentially lose in other places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error
When you see a voter survey result that mentions a margin of error, it's giving you an idea of how precise the survey's results are. Basically, the margin of error tells us how much the survey results could differ from the true situation in the entire population. For instance, in our exercise, the survey indicated Candidate X had 55% support. However, the margin of error is 2%.
This means the actual support could realistically be 2% higher or lower than reported. The range then becomes anywhere from 53% to 57%.
  • A higher margin of error means less confidence in the survey results.
  • Lower margins of error imply greater precision.
In survey data interpretation, it's crucial to acknowledge the margin of error to understand the potential variation in the actual outcomes.
Sample Percentage
The sample percentage represents the proportion of the survey's sample that expresses a particular view. If 55% of the survey participants say they'll vote for Candidate X, as in our example, this is the sample percentage.
It's a snapshot of the sample survey
  • The sample percentage might not perfectly represent the whole population.
  • It can provide insights into the sample's tendencies.
It's always necessary to consider that this number comes with a margin of error and it connects to the confidence interval, providing a potential range for the true population percentage.
Representative Sampling
Representative sampling aims to ensure that the survey results reflect the views of the entire population, not just a small group that might have similar characteristics. If a sample doesn't represent the broader population well, the survey's findings can be misleading.
In the exercise, the survey comes from New York City voters.
This can be a problem if Candidate X is running for a national position, like the U.S. presidency. New York City might not reflect the entire country's preferences, making it a "non-representative" sample for a nationwide election.
  • A representative sample considers diversity and all subgroups of a population.
  • It minimizes biases in survey results.
Surveys must carefully select participants to create truly representative results that accurately inform us about larger groups.
Voter Survey
Voter surveys are practical tools for gathering opinions and gauging electoral outcomes. They are especially useful when trying to predict how an election might turn out.
But like any tool, they have limitations and important factors to consider.
  • Who's in the survey?
  • Is it representative?
  • What methods were used to collect data?
In our case, the survey showed Candidate X with 55% support from New York City voters. But these results might not hold in other regions.
Understanding the context of where and how the survey was conducted can significantly impact the reading of its results. Surveys are invaluable in predicting general trends, but they require careful execution and interpretation for full accuracy.

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