91Ó°ÊÓ

The 2004 Naples, Florida, mini marathon \((13.1 \text { miles) had } 1228\) registrants (Naples Daily News, January 17,2004 ). Competition was held in six age groups. The following data show the ages for a sample of 40 individuals who participated in the marathon. \\[ \begin{array}{lcccc} 49 & 33 & 40 & 37 & 56 \\ 44 & 46 & 57 & 55 & 32 \\ 50 & 52 & 43 & 64 & 40 \\ 46 & 24 & 30 & 37 & 43 \\ 31 & 43 & 50 & 36 & 61 \\ 27 & 44 & 35 & 31 & 43 \\ 52 & 43 & 66 & 31 & 50 \\ 72 & 26 & 59 & 21 & 47 \end{array} \\] a. Show a stretched stem-and-leaf display. b. What age group had the largest number of runners? c. What age occurred most frequently? d. A Naples Daily News feature article emphasized the number of runners who were "20something." What percentage of the runners were in the 20 -something age group? What do you suppose was the focus of the article?

Step by step solution

01

Organize Data for Stem-and-Leaf Display

To create a stem-and-leaf plot, first sort the ages in ascending order: 21, 24, 26, 27, 30, 31, 31, 31, 32, 33, 35, 36, 37, 37, 40, 40, 43, 43, 43, 43, 44, 44, 46, 46, 47, 49, 50, 50, 50, 52, 52, 55, 56, 57, 59, 61, 64, 66, 72. The tens digit will represent the stem, and the units digit will represent the leaf.

02

Create Stem-and-Leaf Display

Using the sorted data, build a stem-and-leaf display: ``` 2 | 1 4 6 7 3 | 0 1 1 1 2 3 5 6 7 7 4 | 0 0 3 3 3 3 4 4 6 6 7 9 5 | 0 0 0 2 2 5 6 7 9 6 | 1 4 6 7 | 2 ``` The stems are the tens digits (2, 3, 4, etc.), and the leaves are the unit digits.

03

Identify Age Group with Largest Number of Runners

Examine the stem-and-leaf display to determine which stem has the most leaves. The stem '4' represents ages from 40 to 49, which has 12 leaves.

04

Identify Most Frequent Age

Look for leaves that appear multiple times. The number 43 appears four times, making it the mode.

05

Calculate Percentage of 20-something Runners

The 20-something age group includes ages 20 to 29. The ages in this group are 21, 24, 26, and 27, so there are 4 such runners. Calculate the percentage: \[ \frac{4}{40} \times 100 = 10\% \] of the runners are in the '20-something' group.

06

Interpret Focus of the Article

The article likely emphasizes the participation of younger runners to highlight increased interest or setting of trends by the younger demographic in running the marathon.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Data Analysis

Data analysis is a fundamental process used to evaluate datasets and draw meaningful conclusions. In the context of this stem-and-leaf plot exercise, data analysis helps us systematically process the ages of marathon participants. By doing this, we can identify patterns, such as the most common age group.
Data analysis involves:

• Organizing Data: We begin by sorting the ages in ascending order. This step lays the foundation for further analysis and helps present a clearer picture of age distribution.


• Data Presentation: The stem-and-leaf plot is a tool that shows data distribution in an organized way. Stems represent the leading digits, while leaves show the trailing digits. It's akin to a quick view of the data's shape.


• Interpretation: Once the data is organized, we can easily see which age group had the most participants, which age appeared most frequently (mode), and find any outliers or trends.

In essence, data analysis in this exercise provides insights into the demographic profile, making it easier to see who runs and participates the most in the marathon.

Mathematical Statistics

Mathematical statistics is pivotal in understanding and interpreting data with precision. With our stem-and-leaf plot, we're engaging in statistical techniques to make sense of raw data. This method helps in deriving a visual summary of data.

• Central Tendency: Finding the mode, which is 43 in this case, provides insight into the most typical age of participants.


• Data Distribution: Through the stem-and-leaf, we can see clusters of ages. This tells us how ages are spread out or concentrated.
  For instance, the 40-49 age group has the most entries, making it the stem with the most leaves.


• Percentage Calculation: To compute the percentage of a specific age group, like the '20-something' group, we use the formula: \[ \text{Percentage} = \left( \frac{\text{Number in Group}}{\text{Total Number}} \right) \times 100 \] Here, we identified a 10% representation of the 20's age range.

By applying these statistical principles, we can make conclusions about the age distribution in the marathon participants.

Demographic Analysis

Demographic analysis examines the characteristics of a specific population; here, the mini marathon participants based on age. This type of analysis can inform various aspects of planning and interpretation.

• Age Segmentation: By breaking down the participants into age groups (20s, 30s, 40s, etc.), we can identify which segment is the most active. This helps understand trends and which groups engage more in marathon running.


• Age Group Focus: The exercise found the 40-49 age group had the largest number of participants, suggesting middle-aged runners participate more in this event.


• Young Adult Highlight: The article's spotlight on younger runners, especially those in their 20s, likely focuses on emerging trends and increased participation in healthy lifestyles among young adults.

This demographic breakdown provides insight into the interests and behaviors of different age groups, aiding event organizers and marketers to tailor their offerings to attract diverse participants.