/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 How old are professional footbal... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the following ordered data: $$\begin{array}{lllllll} 2 & 5 & 5 & 6 & 7 & 7 & 8 & 9 & 10 \end{array}$$ (a) Find the \(\operatorname{low}, Q_{1},\) median, \(Q_{3},\) high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid \(\mathbf{E}: x\) variable $$\begin{array}{ccccccc} 13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \end{array}$$ Grid H: \(y\) variable $$\begin{array}{lllllll} 11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & \end{array}$$ (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y,\) and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(C V\) s to compare the two grids. If \(s\) represents variability in the signal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(C V\) be better, or at least more exciting? Explain.

Consider the numbers 2 3 4 5 5 (a) Compute the mode, median, and mean. (b) If the numbers represent codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represent one-way mileages for trails to different lakes, which average(s) would make sense? (d) Suppose the numbers represent survey responses from 1 to \(5,\) with \(1=\) disagree strongly, \(2=\) disagree, \(3=\) agree, \(4=\) agree strongly, and \(5=\) agree very strongly. Which averages make sense?

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

Basic Computation: Weighted Average Find the weighted average of a data set where 10 has a weight of \(5 ; 20\) has a weight of \(3 ; 30\) has a weight of 2
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.