In a recent survey, 58% of voters indicated that they voted for the incumbent mayor. This survey is considered accurate within plus or minus 4%, 19 times out of 20.

Question

a. There is a 95% probability that between 53% and 63% of voters voted for the incumbent mayor. b. There is a 95% probability that between 54% and 62% of voters voted for the incumbent mayor. c. There is a 99% probability that between 54% and 62% of voters voted for the incumbent mayor. d. There is a 90% probability that between 54% and 62% of voters voted for the incumbent mayor.

Answer 1

The range of percentages is 58-4=54% to 58+4=62%. If there were N voters, then 0.62N-0.54N= 0.08N constitutes 19/20=95% of the voters for the incumbent mayor. For example, if there were 95,000 voters, then between 51,300 and 58,900 voters (7600 voters) voted for the incumbent mayor. So 7,600 voters would represent 95% of those voting for the incumbent mayor, making the total number of such voters 100×7600÷95=8,000 voters. The likelihood is that 400 voters who voted for the incumbent mayor have been excluded from the count. Answer b seems to fit the situation.