Ages of Prisoners The accompanying frequency distribution summarizes sample data consisting of ages of randomly selected inmates in federal prisons (based on data from the Federal Bureau of Prisons). Use the data to construct a 95% confidence interval estimate of the mean age of all inmates in federal prisons.

Verified step by step guidance

1. Step 1: Calculate the midpoint for each age group. The midpoint is the average of the lower and upper bounds of each age group. For example, for the age group 16–25, the midpoint is (16 + 25) / 2 = 20.5.
2. Step 2: Multiply the midpoint of each age group by the corresponding frequency (number of inmates) to find the weighted contribution of each group to the total sum. For example, for the age group 16–25, the contribution is 20.5 × 13.
3. Step 3: Sum up all the weighted contributions from Step 2 to calculate the total sum of ages. Also, sum up all the frequencies to find the total number of inmates.
4. Step 4: Divide the total sum of ages by the total number of inmates to calculate the sample mean age. This is the point estimate for the mean age.
5. Step 5: Use the formula for the standard error of the mean (SE = s / √n, where s is the sample standard deviation and n is the sample size) and the t-distribution to construct the 95% confidence interval. The confidence interval is given by: Mean ± (t * SE), where t is the critical value from the t-distribution table for 95% confidence and degrees of freedom (df = n - 1).

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true mean. This concept is crucial for estimating population parameters based on sample data.

Mean

The mean, or average, is a measure of central tendency that is calculated by summing all values in a dataset and dividing by the number of values. In the context of the ages of prisoners, the mean age provides a single value that represents the central point of the age distribution, which is essential for constructing the confidence interval.

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. In this case, the table shows the number of prisoners within specific age ranges. Understanding frequency distributions is important for analyzing data, as it helps identify patterns and informs the calculation of statistics like the mean and confidence intervals.