In a survey of 2230 U.S. adults, 1272 think that air travel...

In a survey of 2230 U.S. adults, 1272 think that air travel is much more reliable than taking cruises. Construct a $95 \%$ confidence interval for the population proportion of U.S. adults who think that air travel is much more reliable than taking cruises.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. It provides an estimated range of plausible values for the parameter and is associated with a confidence level, such as 95%, indicating the degree of certainty in the estimation process.

Population Proportion

The population proportion refers to the fraction or percentage of individuals within the entire population who possess a particular characteristic or attribute. Since the true population proportion is usually unknown, it is estimated using sample data.

Sample Proportion

The sample proportion is the ratio of individuals in a sample who exhibit a certain characteristic to the total number of individuals in that sample. It serves as an unbiased estimator for the population proportion when the sample is representative.

Standard Error

The standard error of a statistic, such as a sample proportion, measures the variability of the estimator across different samples drawn from the same population. It quantifies the amount of uncertainty in the estimation of the population parameter.

Margin of Error

The margin of error is the product of the critical value and the standard error, and it represents the maximum expected difference between the sample statistic and the true population parameter at a given confidence level. It defines the width of the confidence interval.

Critical Value

The critical value is a threshold derived from a probability distribution that corresponds to the desired confidence level. For example, for a 95% confidence level, the critical value is typically about 1.96 when using the normal distribution, reflecting the number of standard errors to extend on both sides of the sample estimate.

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