Final answer:
A confidence interval offers a range of values, calculated from the sample data, that likely includes the population parameter. At a specified confidence level, it indicates the frequency with which we expect the interval to contain the true population proportion. Higher confidence levels generally require larger sample sizes or greater margins of error.
Step-by-step explanation:
Understanding Confidence Intervals
When a study estimates that 58% of the registers of a town favor expanding the park with an error range of plus or minus 8% points at 95% confidence, we are dealing with the calculation of a confidence interval for the proportion of the population that supports a specific opinion.
A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. For example, if nine of 48 respondents rated the likelihood of a certain event occurring as likely, and we apply the plus-four method to create a confidence interval, we would add four to both the number of successes and failures, then calculate the new proportion and use this to find the confidence interval at a specified confidence level.
When we say we are 95% confident, it means that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true population proportion. So, when we estimate with a 90 percent confidence level that a certain proportion of students are registered voters, it implies that there is a 90% chance that the true population proportion falls within our calculated range.
When calculating confidence intervals, if we need a higher degree of confidence, we must either accept a larger margin of error or increase our sample size. For instance, to be at 96 percent confidence, we have to survey more people or expand the error bound if we aim to estimate the population mean length of time to within one hour.