Inmates In 2002, the mean age of an inmate on death row was...

Inmates In $2002,$ the mean age of an inmate on death row was 40.7 years, according to data obtained 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 deathrow inmates and finds that their mean age is $38.9,$ with a standard deviation of $9.6 .$ (a) Do you believe the mean age has changed? Use the $\alpha=0.05$ level of significance. (b) Construct a $95 \%$ confidence interval about the mean age. What does the interval imply?

Inmates In $2002,$ the mean age of an inmate on death row was 40.7 years, according to data obtained 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 deathrow inmates and finds that their mean age is $38.9,$ with a standard deviation of $9.6 .$
(a) Do you believe the mean age has changed? Use the $\alpha=0.05$ level of significance.
(b) Construct a $95 \%$ confidence interval about the mean
age. What does the interval imply?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. In the context of comparing means, a null hypothesis is established stating that there is no difference or change, and an alternative hypothesis is posed suggesting that there is a change.

One-Sample t-Test

A one-sample t-test is used when comparing the sample mean to a known or hypothesized population mean, particularly when the population standard deviation is unknown. It relies on the t-distribution, which accounts for additional uncertainty from estimating the standard deviation from a small sample.

Significance Level (?)

The significance level, commonly denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of a Type I error, which is rejecting the null hypothesis when it is actually true. Commonly, a significance level of 0.05 is used, indicating a 5% risk of error.

t-Distribution and Degrees of Freedom

The t-distribution is used in constructing test statistics and confidence intervals when the sample size is small and the population standard deviation is unknown. Degrees of freedom, generally calculated as the sample size minus one, reflect the amount of independent information available for estimating the population parameter.

Confidence Interval

A confidence interval provides a range of values which, with a certain level of confidence (such as 95%), is believed to contain the true population mean. It is constructed using the sample mean, the appropriate critical value from the t-distribution, and the standard error, which reflects the variability of the sample mean.

Standard Error and Margin of Error

The standard error of the mean quantifies the variability of the sample mean from the true population mean, calculated as the sample standard deviation divided by the square root of the sample size. The margin of error is the product of the critical value and the standard error, defining the range above and below the sample mean that forms the confidence interval.

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