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Chapter 3: Problem 22

How old are professional football players? The 11th edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years: $$\begin{array}{llllllllll}24 & 23 & 25 & 23 & 30 & 29 & 28 & 26 & 33 & 29 \\\24 & 37 & 25 & 23 & 22 & 27 & 28 & 25 & 31 & 29 \\\25 & 22 & 31 & 29 & 22 & 28 & 27 & 26 & 23 & 21 \\\25 & 21 & 25 & 24 & 22 & 26 & 25 & 32 & 26 & 29\end{array}$$ (a) Compute the mean, median, and mode of the ages. (b) Interpretation Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.

Short Answer

Expert verified
Mean: 25.64, Median: 25, Mode: 25. Mode and median closely represent players' ages, more than mean due to fewer older outliers.

Step by step solution

01

Organize Data

List the ages in increasing order to easily identify the median and mode: 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 31, 31, 32, 33, 37.
02

Calculate the Mean

The mean is calculated by summing all ages and dividing by the number of players. Sum of ages is 923 and there are 36 players. Mean = \( \frac{923}{36} \approx 25.64 \).
03

Determine the Median

With 36 values, the median is the average of the 18th and 19th values in our ordered list. Both are 25, so the median is 25.
04

Identify the Mode

The mode is the number that appears most frequently. Age 25 appears 6 times, more than any other age. Thus, the mode is 25.
05

Interpret Results

The mean is 25.64, median is 25, and mode is 25. The mode and median suggest a central tendency around 25, while the mean is slightly higher because of a few older ages such as 37. Overall, ages concentrate around 25.

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Key Concepts

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

Mean Calculation
The mean is a central value that gives us an idea of the average age of the football players in this exercise. To find the mean, we add up all the ages and then divide by the number of players. Here is the step-by-step process:
  • Step 1: Sum the Ages - Add all the ages together: 21 + 21 + 22 + ... + 37 = 923.
  • Step 2: Count the Players - There are 36 players in total.
  • Step 3: Divide - Divide the total sum by the number of players: \( \frac{923}{36} \approx 25.64 \).
The mean age is approximately 25.64. This gives us a general idea of the average age of players, but it can be influenced by unusually high or low ages.
Median Determination
Determining the median is one way to find the middle value in a list of numbers. It helps us understand the age around which most players are grouped. To find the median for our data:
  • Step 1: Order the Ages - Arrange the ages in increasing order: 21, 21, 22,..., 37.
  • Step 2: Locate the Middle - Since there are 36 ages, the median will be the average of the 18th and 19th number in this ordered list.
  • Step 3: Calculate the Median - Both the 18th and 19th ages are 25, so the median is 25.
The median age is 25, which indicates that half of the players are younger than or equal to 25, and the other half is older. This value is less affected by outliers like the age 37.
Mode Identification
The mode identifies the most common age among the players. It shows which age appears more frequently than others. Here's how to find it:
  • Step 1: Frequency Count - Count the number of times each age appears in the ordered list.
  • Step 2: Identify Most Frequent - Determine which age appears the most. Age 25 appears 6 times.
Thus, the mode is 25. This suggests that age 25 is the most typical age among the football players. Unlike the mean or median, the mode is not influenced by outlier ages.

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