Use the technology of your choice to find the required...

Use the technology of your choice to find the required confidence interval. In a nationwide survey of U.S. adults by the Cincinnati-based research firm Directions Research Inc., only 276 of the 1063 respondents said they would purchase or lease a new car from a manufacturer that had declared bankruptcy. Determine and interpret a $90 \%$ confidence interval for the percentage of all U.S. adults who would purchase or lease a new car from a manufacturer that had declared bankruptcy.

Use the technology of your choice to find the required confidence interval.
In a nationwide survey of U.S. adults by the Cincinnati-based research firm Directions Research Inc., only 276 of the 1063 respondents said they would purchase or lease a new car from a manufacturer that had declared bankruptcy. Determine and interpret a $90 \%$ confidence interval for the percentage of all U.S. adults who would purchase or lease a new car from a manufacturer that had declared bankruptcy.

Key Concepts

Interpretation of a Confidence Interval

Interpreting a confidence interval means understanding that, if the same sampling process were repeated numerous times, a certain percentage (the confidence level) of the computed intervals would contain the true population parameter. It provides a probabilistic statement about the reliability of the estimate.

Z-Score (Critical Value)

The Z-score is a statistical measure that corresponds to the specified confidence level in the standard normal distribution. It is used to determine the margin of error when constructing a confidence interval for a proportion.

Margin of Error

The margin of error is the product of the critical value (from the normal or t-distribution) and the standard error. It represents the maximum expected difference between the true population proportion and the sample proportion, given a certain confidence level.

Standard Error

The standard error measures the variability of the sample proportion estimate from sample to sample. It is calculated using the formula ?[p(1-p)/n], where p is the sample proportion and n is the sample size, providing insight into the precision of the estimate.

Sample Proportion

The sample proportion is the ratio of the number of observed successes to the total number of observations. It serves as an unbiased point estimate for the unknown true probability or proportion in the population.

Confidence Interval

A confidence interval provides a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. It quantifies the uncertainty inherent in using sample data to estimate a population characteristic.

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