Question

To examine the effect of exercise on body composition, data were collected from a sample of active women who engaged in 9 hours or more of physical activity per week ("Effects of Habitual Physical Activity on the Resting Metabolic Rates and Body Composition of Women Aged 35 to 50 Years," Journal of the American Dietetic Association [2001]: 1181-1191). Percent of body fat was measured for each woman. The researchers planned to use the resulting data to learn about mean percent body fat for active women.

Short answer

The mean percent body fat for active women can be calculated using the arithmetic mean formula. It requires the total sum of all body fat percentages of the active women and the total number of women in the sample for the calculation.

Step-by-step solution

Collect data of body fat percentages

Start by gathering the data of the percentages of body fat from the active women who engaged in 9 hours or more of physical activity per week. This data might be provided in a chart or a list.

Sum up all the percentages

The next step is to sum up all the percentages of body fat. You could do this by using a calculator or a computer program capable of processing numerical data.

Count the number of data entries

Count the total number of active women in the sample. This number is needed in order to calculate the mean of the body fat percentages.

Calculate the mean percent body fat

Now, to obtain the mean body fat percentage, divide the total sum of body fat percentages by the number of data entries. The formula for calculating the arithmetic mean is given as \( Mean = \frac{\text{Sum of all data points}}{\text{Total number of data points}} \). Apply this formula to calculate the mean body fat percentage for active women.

Key concepts

Body Fat Percentage

Understanding the concept of body fat percentage is fundamental when studying body composition. Body fat percentage is a measure that represents the proportion of fat one has in their body, as compared to their total mass. It's a vital health and fitness metric, distinct for each individual, and can vary widely based on factors like age, sex, genetics, and level of physical activity.

It is calculated by dividing the weight of a person's fat by their total weight and then multiplying the result by 100 to get a percentage value. In the context of research or health assessments, accurately measuring body fat percentage helps to evaluate risks associated with obesity or to monitor changes in body composition over time as a result of lifestyle changes, such as increased physical activity.

For instance, the study on active women aims to determine an average body fat percentage by collecting individual measurements. The goal is to provide insights into the typical body composition of a specific, physically active demographic, which in turn could inform health guidelines or fitness strategies.

Mean Calculation

Mean calculation is a fundamental concept not only in statistics but also in everyday life, as it helps to summarize a large set of numbers with a single value. The mean, commonly known as the average, is calculated by adding all the numerical values together and then dividing by the count of those values.

To illustrate, consider a set of body fat percentage data collected from multiple individuals. If the percentages are 18%, 22%, 20%, and 19%, the mean body fat percentage would be calculated as follows: \( Mean = \frac{18 + 22 + 20 + 19}{4} \), which equals 19.75%. This result is the average body fat percentage for the group. In research studies like the one mentioned, the mean gives a central value around which individual data points can vary, offering a snapshot of the typical or expected value in the sample population.

Data Collection in Statistics

Data collection is a critical process in the field of statistics, wherein data is gathered to provide information about a specific subject. This process includes planning the data collection in terms of what to gather and how to accurately and consistently measure it.

In the context of measuring body fat percentage in active women, researchers would first define the population of interest and select an appropriate sample size. They would then choose reliable methods to measure body fat, such as calipers for skinfold measurement, bioelectrical impedance analysis, or dual-energy X-ray absorptiometry (DEXA).

It's essential during data collection to ensure accuracy and precision to make the findings meaningful; otherwise, the statistical analysis may lead to incorrect conclusions. In our exercise instance, a consistent method of body fat measurement across all participants is crucial to obtain valid results that reflect the true mean body fat percentage in the studied population.