Age of Death-Row Inmates In 2002, the mean age of an inmate...

Age of Death-Row Inmates In $2002,$ the mean age of an inmate on death row was 40.7 years, according to data from the U.S. Department of Justice. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selects 32 death-row inmates and finds that their mean age is 38.9 with a standard deviation of 9.6. Construct a $95 \%$ confidence interval about the mean age. What does the interval imply?

Key Concepts

Interpretation of Confidence Interval

Interpreting a confidence interval involves understanding that, under repeated sampling, a given percentage (e.g., 95%) of such intervals would contain the true population parameter. This does not mean there is a 95% probability that any one interval contains the parameter; rather, it reflects the long-run performance of the interval estimation method.

Population Parameter Estimation

Population parameter estimation involves using sample data to infer characteristics of the entire population, such as the mean age in this context. Confidence intervals are a fundamental tool in this process, offering a way to express the precision and uncertainty associated with the estimate.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. In the context of estimating the mean of a population, the interval provides a level of assurance (such as 95%) that the interval covers the true mean, based on the variability in the sample and the chosen confidence level.

t-Distribution

The t-distribution is used instead of the normal distribution when dealing with small sample sizes or when the population standard deviation is unknown. It accounts for the extra uncertainty that comes from estimating the standard deviation from a sample, and its shape depends on the degrees of freedom, which in turn are related to the sample size.

Standard Error

The standard error is a measure of the variability or dispersion of the sample mean from the population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size, indicating how much the sample mean is expected to vary from sample to sample.

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