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The following data give the one-way commuting times (in minutes) from home to work for a random sample of 50 workers. $$ \begin{array}{llllllllll} 23 & 17 & 34 & 26 & 18 & 33 & 46 & 42 & 12 & 37 \\ 44 & 15 & 22 & 19 & 28 & 32 & 18 & 39 & 40 & 48 \\ 16 & 11 & 9 & 24 & 18 & 26 & 31 & 7 & 30 & 15 \\ 18 & 22 & 29 & 32 & 30 & 21 & 19 & 14 & 26 & 37 \\ 25 & 36 & 23 & 39 & 42 & 46 & 29 & 17 & 24 & 31 \end{array} $$ a. Construct a frequency distribution table using the classes \(0-9\), \(10-19,20-29,30-39\), and \(40-49 .\) b. Calculate the relative frequency and percentage for each class. c. Construct a histogram for the percentage distribution made in part b. d. What percentage of the workers in this sample commute for 30 minutes or more? e. Prepare the cumulative frequency, cumulative relative frequency, and cumulative percentage distributions using the table of part a.

Step by step solution

01

- Create a Frequency Distribution Table

Firstly, separate the dataset into the classes given in the problem: \(0-9, 10-19, 20-29, 30-39\), and \(40-49\). Count the number of data points in each class to find the frequency. The frequency of each class is nothing but the count of data points within that class.

02

- Compute Relative Frequency and Percentage

Next, calculate the relative frequency and percentage for each class. The relative frequency is the frequency of a class divided by the total number of data points. Multiply the relative frequency by 100 to get the percentage for each class.

03

- Plot a Histogram

Taking the data from Step 2, plot a histogram with the classes on the x-axis and the percentage on the y-axis.

04

- Calculate Commuting Time Percentage

To find the percentage of workers who commute for 30 minutes or more, sum the frequency of the classes equal to or over 30 minutes. Then, divide by the total number of workers and multiply by 100 to get the percentage.

05

- Compute Cumulative Frequency, Relative Frequency, and Percentage

Finally, calculate the cumulative frequency, cumulative relative frequency, and cumulative percentage. The cumulative frequency is the sum of the frequencies up to a certain class. The cumulative relative frequency is the cumulative frequency divided by the total number of data points. Multiply the cumulative relative frequency by 100 to get the cumulative percentage. This can be visualized with a cumulative frequency graph.

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Key Concepts

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

Understanding Relative Frequency

Relative frequency is a simple concept that helps us understand how often a certain event occurs relative to the total number of events. In our exercise, this refers to the frequency of workers falling into a commuting time class divided by the total number of workers in the sample. For instance, if a class has a frequency of 10, and there are 50 workers in total, the relative frequency would be \( \frac{10}{50} = 0.2 \).

• This simple division helps us compare frequencies across different classes, providing a normalized measure that isn't dependent on the total size of the dataset.
• The relative frequency is also foundational in creating other distributions, like percentage distribution.

Once you have calculated the relative frequency for each class, multiplying it by 100 will give you the percentage of workers in that class, offering an even clearer comparison.

Creating and Interpreting a Histogram

A histogram is a type of bar graph that visualizes the distribution of numeric data. In our exercise, we use a histogram to represent the percentage distribution of commuting times. The x-axis lists the classes of commuting times (e.g., \(0-9, 10-19\)), while the y-axis represents the percentage of observations in each class.

• Each bar's height corresponds to the percentage of workers with commuting times within that class range.
• This visualization makes it easy to spot trends, such as which commuting time ranges are most common.
• Interpreting a histogram can show you where most of your data points lie and if there's any skewness in the dataset.

For example, if most bars are high in the \(20-29\) and \(30-39\) classes, it suggests that most workers commute around 20 to 39 minutes.

Exploring Cumulative Frequency

Cumulative frequency involves adding up the frequencies of each class, starting from the smallest class and going to the largest. This gives you a running total of frequencies, showing how many data points fall below or at a particular class boundary.

• The key idea here is accumulation: as you move from class to class, you accumulate the counts from previous classes as well.
• It's very useful in identifying medians and quartiles, as you can see at a glance how many data points are below a given value.

For instance, if the cumulative frequency for the \(30-39\) class is 40, that means 40 workers have a commuting time of 39 minutes or less. Calculating cumulative relative frequency and cumulative percentage follows a similar pattern, but instead of the raw frequency count, you use the relative frequencies to paint a proportional picture.

Understanding Percentage Distribution

Percentage distribution is simply a way of expressing relative frequencies as percentages. By converting relative frequencies into percentages, it becomes easier to understand and communicate the data, as percentages are more intuitive for most people.

• To obtain this, multiply the relative frequency by 100 for each class.
• Having data in percentage form helps in quick comparisons, showing proportions more visibly.

For example, if the relative frequency of a class is 0.3, its percentage distribution is 30%. This makes it straightforward to convey that 30% of workers fall into that commuting time range, making it a powerful tool for analysis and reporting.