Inference from sample statistics and margin of error：練習問題

A recent poll found that \( 45\% \) of a city's residents support a new park proposal, with a margin of error of \( \pm 3\% \). Which of the following is the most appropriate interpretation of this margin of error?

A poll of 1,000 likely voters found that 52% favor a proposed new law, with a margin of error of \( \pm 3\% \) at a 95% confidence level. Which of the following statements is best supported by this data?

A researcher conducted a survey with a random sample of 400 people and found a margin of error of \( \pm 4\% \). If the researcher wants to reduce the margin of error to \( \pm 2\% \) while keeping the same confidence level, which of the following is the most appropriate new sample size?

A statistical survey conducted with a random sample of \( n \) people produced a margin of error of \( 6\% \). If the researcher wants to reduce the margin of error to \( 3\% \), what should be the new sample size in terms of \( n \), assuming the sample proportion remains similar?

A study reports a \( 95\% \) confidence interval for the mean height of a population as \( (160, 170) \) centimeters based on a sample of size \( N \). If a new sample of size \( 4N \) is taken from the same population and results in the same sample mean and standard deviation, what would be the new \( 95\% \) confidence interval?

A polling organization conducted a survey of a random sample of \( 1,200 \) registered voters in a city with a total population of \( 250,000 \) registered voters. The survey found that \( 54\% \) of the respondents plan to vote for Candidate A, with a margin of error of \( \pm 3\% \) at a \( 95\% \) confidence level.

Part A: Based on this survey, what is the range of the estimated number of registered voters in the entire city who plan to vote for Candidate A?
Part B: If the organization wanted to reduce the margin of error to \( \pm 1.5\% \) while maintaining the same confidence level, should they increase or decrease the sample size? Explain the relationship between sample size and margin of error.