Question

For each of the following statements, identify the number that appears in boldface type as the value of either a population characteristic or a statistic: a. A department store reports that \(84 \%\) of all customers who use the store's credit plan pay their bills on time. b. A sample of 100 students at a large university had a mean age of \(24.1\) years. c. The Department of Motor Vehicles reports that \(22 \%\) of all vehicles registered in a particular state are imports. d. A hospital reports that based on the 10 most recent cases, the mean length of stay for surgical patients is \(6.4\) days. e. A consumer group, after testing 100 batteries of a certain brand, reported an average life of \(\mathbf{6 3} \mathrm{hr}\) of use.

Short answer

a. Population characteristic, b. Statistic, c. Population characteristic, d. Statistic, e. Statistic

Step-by-step solution

Statement a Analysis

Here, the percentage that \(84\%\) of all customers who use the store's credit plan pay their bills on time refers to the population. So, it is a population characteristic.

Statement b Analysis

In this statement, the information about the mean age of \(24.1\) years is derived from a sample of 100 students at a large university. Hence, this number is a statistic.

Statement c Analysis

In this scenario, the number \(22\%\) refers to all vehicles in a specific state. Thus, it's a population characteristic.

Statement d Analysis

This statement gives the mean length of stay for surgical patients based on the 10 most recent cases. Since it's based on a sample (10 most recent cases), this is a statistic.

Statement e Analysis

The average life of a battery is tested on a sample of 100 batteries. Hence, this is a statistic.

Key concepts

Population Characteristic

A population characteristic is a parameter that defines an aspect of the entire population. It could be a number or a trait that doesn't change regardless of the group you examine within that population. For example, in the exercise provided, the statement about the department store reporting that 84% of all customers who use the store's credit plan paying their bills on time is a population characteristic. This figure is understood to represent the behavior of all customers, not just a selected group.

In essence, a population characteristic could be a proportion, such as the 22% of all vehicles in a certain state being imports, or it could be a mean or median that describes something about everyone or everything in the group being considered. These are considered constants in statistical analyses and serve as benchmarks against which to compare sample statistics.

Statistic

On the other hand, a statistic refers to a measure derived from a sample, which is a subset of the population. In our textbook exercise, the mean age of 24.1 years from a sample of 100 students is an example of a statistic, as it pertains to the characteristics of a sample rather than the whole population. Likewise, the mean length of stay for surgical patients reported as 6.4 days based on the 10 most recent cases is also a statistic.

Statistics are crucial in inferential statistics, where we draw conclusions about the population based on sample data. However, they are always subject to a margin of error and variability because they are calculated from a subset, not the entire group.

Mean

The mean is one of the central measures of central tendency in statistics. It is calculated by summing all the values in a data set and dividing by the number of values. For example, if a student group has ages of 23, 24, 27, and 19, their mean age is \( (23+24+27+19) / 4 = 93 / 4 = 23.25 \) years. From the exercise, the sample mean age of 24.1 years for students represents such an average, though calculated from a sample, not the entire student body.

The mean provides a quick snapshot of the 'average' but can be skewed by extremely high or low values. Thus, it's an essential descriptor in statistics but must be used judiciously.

Percentage

Percentages are used in statistics to represent how a part compares to a whole, always relative to a base of 100. They are a way to standardize comparisons and are widely used because they are easy to understand. In both reported population characteristics and calculated statistics in our exercise, percentages manifest as 84% of customers paying on time and 22% of all vehicles being imports. Understanding percentages is crucial because they allow us to quantify characteristics and statistics in an immediately graspable way, whether we're referring to a whole population or a sample.

Sample

A sample is a subset of a population selected to represent the whole. It is used in statistics because it's often impractical or impossible to gather data from an entire population. In the given scenarios, a group of 100 students and the 10 most recent surgical cases serve as samples from which statistics like mean age and mean length of stay are calculated.

When taking a sample, it's important to ensure that it's representative, meaning it accurately reflects the larger population's characteristics. This is often done through random selection. An unrepresentative sample can lead to biased statistics and faulty conclusions about the population.