The correct answer is:

There is a 95% probability that the proportion of voters that will support the ballot is between 48% and 56%.

This statement reflects the margin of error and the confidence interval typically reported in polling. The margin of error is an estimate of the range in which the true population proportion (the percentage of voters who support the proposition) is likely to lie, based on the sample.

Here's a breakdown of why this is the correct answer:

• The polling agency found that 52% of those surveyed support Prop XYZ, and the margin of error is 4%.
• This means the true proportion of voters in the population is likely to be within the range of 48% to 56%, with a certain level of confidence (usually 95%).
• "95% probability" is key here because it represents the confidence level, meaning that if the poll were repeated many times, 95% of those samples would produce results within this range of 48% to 56%.

Why the other options are not correct:

1. "There is a 95% probability that another sample of voters will have between 48% and 56% supporting XYZ."

   • This refers to the likelihood of a specific sample having results within that range, but what you're really describing is the population parameter's range, not the sample range.

2. "The proportion of voters that support Prop XYZ is between 48% and 56% with the agency's chosen level of confidence."

   • This phrasing is technically misleading. We cannot know for certain the true population proportion is within this range. It’s about confidence based on the sample data, not certainty.

3. "With 95% confidence, Prop XYZ has 52% of voter support."

   • The percentage of 52% refers to the sample result, not the true population value. The confidence level does not guarantee that the true proportion is exactly 52%, only that it is likely within the range of 48% to 56%.

Summary:

The correct interpretation is that the polling agency’s survey suggests with 95% confidence, the true proportion of voters who support Prop XYZ falls between 48% and 56%.

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Would you like more details about margin of error or confidence intervals? Feel free to ask any questions!

Follow-up questions:

1. What is the significance of the margin of error in polling data?
2. How is the margin of error calculated in a survey?
3. What is the meaning of "confidence level" in statistics?
4. How do polling agencies ensure the sample is representative of the population?
5. What factors could increase or decrease the margin of error in a poll?

Tip: In surveys, increasing the sample size can reduce the margin of error, making the results more reliable.