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Chapter 2: Problem 28

The following data give the number of turnovers (fumbles and interceptions) made by both teams in each of the football games played by North Carolina State University during the 2009 and 2010 seasons. $$ \begin{array}{llllllllllll} 2 & 3 & 1 & 1 & 6 & 5 & 3 & 5 & 5 & 1 & 5 & 2 \\ 5 & 3 & 4 & 4 & 5 & 8 & 4 & 5 & 2 & 2 & 2 & 6 \end{array} $$ a. Construct a frequency distribution table for these data using single-valued classes. b. Calculate the relative frequency and percentage for each class. c. What is the relative frequency of games in which there were 4 or 5 turnovers? d. Draw a bar graph for the frequency distribution of part a.

Short Answer

Expert verified
Using the frequency distribution table, the team had turnovers most frequently of 2 and 5 times per game. In terms of relative frequency, 41.67% of games had 4 or 5 turnovers. This is represented visually in the bar graph.

Step by step solution

01

Construct a frequency distribution table

Count the frequency of each number from 1 to 8 because these are the values in the data. The frequencies are as follows: \n 1: 3 times \n 2: 5 times \n 3: 3 times \n 4: 4 times \n 5: 6 times \n 6: 3 times \n 8: 1 time\n There are total 24 numbers. This represents the total number of games.
02

Calculate the relative frequency and percentage

The relative frequency is calculated by dividing the frequency of each class (number of turnovers made) by the total number of data points.\n For example, the relative frequency of 1 turnover is \(\frac{3}{24} = 0.125\). This means that the team had 1 turnover in 12.5% of the games. Do this for each class.\n The percentage is calculated by multiplying the relative frequency by 100. For example, for 1 turnover, the percentage is \(0.125 * 100 = 12.5%\). Do this calculation for each class.
03

Calculate the relative frequency of games with 4 or 5 turnovers

Add the relative frequencies of 4 turnovers and 5 turnovers which is \(\frac{4}{24} + \frac{6}{24} = 0.4167\). This means that 41.67% of games had 4 or 5 turnovers.
04

Create a bar graph

Use the frequency distribution table from Step 1 to make the bar graph. Set the numbers of turnovers on the x-axis and the frequencies on the y-axis. Draw bars for each class height proportional to its frequency. This will visually represent the distribution of turnovers per game.

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Key Concepts

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

Frequency Distribution
In statistics, a frequency distribution is a way to organize data into specific intervals, known as classes, and shows how often each value occurs. It offers a visual snapshot of a dataset's structure, making it easier to interpret.

To construct a frequency distribution, follow these steps:
  • List all possible values (or classes) that the data can take. In our exercise, the classes are the numbers of turnovers ranging from 1 to 8.
  • Count how frequently each value appears in the data set. For instance, the number 1 appears 3 times, and the number 5 appears 6 times.
  • Create a table listing each value alongside its frequency. This table represents the frequency distribution.
A frequency distribution breaks the dataset down into manageable parts, letting us see the most and least common values at a glance, which helps in identifying trends.
Relative Frequency
Relative frequency is the fraction of times a particular value occurs compared to the total number of data points. It gives more insight into how significant a certain value is within the dataset.

To calculate relative frequency:
  • Take the frequency of a specific data value. For example, if the number 5 appears 6 times, its frequency is 6.
  • Divide this frequency by the total number of data points, which in our exercise is 24.
  • The result is the relative frequency of that value. So, the relative frequency of 5 turnovers is \(\frac{6}{24} = 0.25\).
Relative frequency is often expressed as a decimal or fraction, but it can be difficult to interpret at a glance without context. This is where percentage calculation can provide more straightforward understanding.
Percentage Calculation
Percentage calculation converts relative frequency into a more comprehensible format for many: a percentage. By representing relative frequencies as percentages, you can easily compare different classes.

Here's how to calculate the percentage:
  • Take the relative frequency of the class. For example, the relative frequency of 1 turnover is \(0.125\).
  • Multiply the relative frequency by 100 to convert it into a percentage. So, for 1 turnover, the percentage becomes \(0.125 \times 100 = 12.5\%\).
Percentages allow us to quickly see the proportion of a class within a dataset in a more intuitive way, making it easy to compare and communicate results.
Bar Graph Construction
Bar graphs are visual representations of data that display the distribution of values across different categories. They are particularly useful for showing frequency distributions.

To create a bar graph:
  • Draw two axes. Typically, the horizontal (x-axis) represents the classes (values of turnovers), while the vertical (y-axis) represents the frequencies.
  • For each class, draw a bar whose height corresponds to its frequency. A higher frequency results in a taller bar.
  • Label each axis clearly, so it's easy to see the relationship between classes and their frequencies.
Bar graphs make it simple to compare classes and can quickly indicate patterns and outliers in data. They鈥檙e an effective tool for summarizing complex datasets in an accessible format.

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Most popular questions from this chapter

The following data give the waiting times (in minutes) for 25 students at the Student Health Center of a university. \(\begin{array}{lllllllllllll}39 & 19 & 19 & 35 & 18 & 32 & 20 & 15 & 20 & 29 & 25 & 32 & 28 \\ 19 & 42 & 18 & 32 & 31 & 21 & 46 & 27 & 13 & 14 & 15 & 28 & \end{array}\) Create a dotplot for these data.

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A July 2011 ESPN SportsNation poll asked, "Which is the best Fourth of July weekend sports tradition?" (http://espn.go.com/espn/fp/flashPollResultsState?sportIndex=frontpage\&pollid=116290). The choices were Major League baseball game (B), Nathan's Famous International Hot Dog Eating Contest (H), Breakfast at Wimbledon (W), or NASCAR race at Daytona (N). The following data represent the responses of a random sample of 45 persons who were asked the same question. \(\begin{array}{lllllllll}\text { H } & \text { H } & \text { B } & \text { W } & \text { N } & \text { B } & \text { H } & \text { N } & \text { W } \\\ \text { N } & \text { H } & \text { B } & \text { W } & \text { H } & \text { N } & \text { N } & \text { H } & \text { H } \\ \text { B } & \text { B } & \text { W } & \text { H } & \text { H } & \text { B } & \text { W } & \text { H } & \text { B } \\ \text { H } & \text { B } & \text { B } & \text { H } & \text { B } & \text { H } & \text { B } & \text { N } & \text { H } \\ \text { B } & \text { B } & \text { H } & \text { H } & \text { H } & \text { B } & \text { H } & \text { H } & \text { N }\end{array}\) a. Prepare a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. What percentage of the respondents mentioned Major League baseball game or Breakfast at Wimbledon? d. Draw a bar graph for the frequency distribution.

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