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Chapter 2: Problem 82

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.

Short Answer

Expert verified
The dotplot should have dots stacked on the numbers 13 through 46 on the x-axis. The height of the stacked dots show the frequency of each waiting time. In other words, the more students that had to wait a certain amount of time, the more dots at that number.

Step by step solution

01

Arrange the data

Arrange the waiting times in ascending order for ease of marking the dots. The data becomes as follows: \[ \begin{array}{lllllllllllll}13 & 14 & 15 & 15 & 18 & 18 & 19 & 19 & 19 & 20 & 20 & 21 & 25 \ 27 & 28 & 28 & 29 & 31 & 32 & 32 & 35 & 39 & 42 & 46 \end{array}\]
02

Prepare the Dotplot

A dot plot generally consists of a horizontal number line (x-axis) along which the data values are aligned. Mark the numbers 13-46 as they represent the waiting time in minutes on the x-axis. These will serve as the positions to place the dots.
03

Plot the data

Now you can start plotting the data. For every waiting time in the data, place a dot above the corresponding time on the number line. If there is more than one occurrence of a waiting time, the dots are stacked above each other.

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Key Concepts

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

Statistics
Statistics is all about collecting, organizing, analyzing, and interpreting data to make informed decisions. It's like being a detective, using numbers to understand the world around us. When dealing with a collection of data like waiting times at a university health center, statistics help in deriving meaningful insights.

In practice, the statistical process involves several steps:
  • Collecting data: Gathering relevant data points, such as waiting times from 25 students.
  • Organizing data: Arranging the data in a logical order, like sorting it from least to greatest.
  • Analyzing data: Looking for patterns or tendencies, such as the frequency of certain waiting times.
  • Interpreting data: Drawing conclusions based on the analysis, which can help in planning or making decisions.
By employing statistics, you can spot trends like the most common waiting time or the time that occurs most frequently. This is particularly useful in identifying any systemic issues that need addressing, such as reducing wait times at the health center for better efficiency.
Data Visualization
Data visualization is a way of transforming complex data sets into graphical representations, making it easier to see patterns, trends, or outliers. One of the simplest and most effective data visualization tools is the dot plot, especially for smaller data sets like the one provided.

Creating a dot plot involves:
  • Using a horizontal axis to represent the continuous data values, such as waiting times.
  • Placing a dot above the corresponding value on the axis for each occurrence.
  • Stacking dots vertically if a value occurs more than once.
This method is particularly excellent for seeing the distribution of data points at a glance. For instance, if we visualize the students' waiting times using a dot plot, we'll immediately recognize which times are most frequent. This offers a quick and straightforward way to share insights with others who may not have a statistical background.
Descriptive Statistics
Descriptive statistics provide a summary of data in a way that is manageable and understandable. They help communicate key aspects of the data's distribution and essential characteristics, relying on metrics such as mean, median, and mode.

When assessing the waiting times:
  • The **mean** provides an average waiting time, summing all times and dividing by the number of students.
  • The **median** offers the middle value when the times are ordered, giving a sense of central tendency.
  • The **mode** reveals the most frequently occurring waiting time, highlighting any common delays.
Increasingly, graphical methods like dot plots complement these descriptive techniques by providing a visual representation of our data. Not only do they make the distribution patterns more evident, but they also spark curiosity to dive deeper into analyzing variability, range, or potential anomalies present in the data.

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Most popular questions from this chapter

The following data give the money (in dollars) spent on textbooks by 35 students during the \(2011-12\) academic year. \(\begin{array}{lllllllll}565 & 728 & 870 & 620 & 345 & 868 & 610 & 765 & 550 \\ 845 & 530 & 705 & 490 & 258 & 320 & 505 & 957 & 787 \\ 617 & 721 & 635 & 438 & 575 & 702 & 538 & 720 & 460 \\ 840 & 890 & 560 & 570 & 706 & 430 & 968 & 638 & \end{array}\) a. Prepare a stem-and-leaf display for these data using the last two digits as leaves. b. Condense the stem-and-leaf display by grouping the stems as \(2-4,5-6\), and \(7-9\).

Briefly explain the three decisions that have to be made to group a data set in the form of a frequency distribution table.

The National Highway Traffic Safety Administration collects information on fatal accidents that occur on roads in the United States. Following are the number of fatal motorcycle accidents that occurred in each of South Carolina's 46 counties during the year 2009 (http://www-fars.nhtsa.dot.gov). \(\begin{array}{rrrrrrrrrrr}3 & 28 & 3 & 35 & 3 & 7 & 13 & 38 & 6 & 44 & 11 & 14 \\ 12 & 18 & 17 & 17 & 6 & 20 & 3 & 7 & 29 & 17 & 51 & 12 \\ 5 & 60 & 12 & 18 & 17 & 21 & 14 & 34 & 3 & 12 & 8 & 5 \\ 11 & 29 & 20 & 40 & 3 & 30 & 23 & 5 & 10 & 23 & & \end{array}\) a. Construct a frequency distribution table using the classes \(1-10,11-20,21-30,31-40,41-50\), and \(51-60\) b. Calculate the relative frequency and percentage for each class. c. Construct a histogram and a polygon for the relative frequency distribution of part b. d. What percentage of the counties had between 21 and 40 fatal motorcycle accidents during \(2009 ?\)

As shown in Exercise \(2.89\), back-to-back stem-and-leaf displays can be used to compare the distribution of a variable for two different groups. Consider the following data, which give the alcohol Flying Dog Brewery: \(\begin{array}{lllllllll}4.7 & 4.7 & 4.8 & 5.1 & 5.5 & 5.5 & 5.6 & 6.0 & 7.1 \\\ 7.4 & 7.8 & 8.3 & 8.3 & 9.2 & 9.9 & 10.2 & 11.5 & \end{array}\) Sierra Nevada Brewery: \(\begin{array}{lllllllllllll}4.4 & 5.0 & 5.0 & 5.6 & 5.6 & 5.8 & 5.9 & 5.9 & 6.7 & 6.8 & 6.9 & 7.0 & 9.6\end{array}\) a. Create a back-to-back stem-and-leaf display of these data. Place the Flying Dog Brewery data to the left of the stems. b. What would you consider to be a typical alcohol content of the beers made by each of the two breweries? c. Does one brewery tend to have higher alcohol content in its beers than the other brewery? If so, which one? Explain how you reach this conclusion by using the stem-and-leaf display. d. Do the alcohol content distributions for the two breweries appear to have the same levels of variability? Explain how you reach this conclusion by using the stem-and-leaf display.

A data set on monthly expenditures (rounded to the nearest dollar) incurred on fast food by a sample of 500 households has a minimum value of \(\$ 3\) and a maximum value of \(\$ 147\). Suppose we want to group these data into six classes of equal widths. a. Assuming that we take the lower limit of the first class as \(\$ 1\) and the upper limit of the sixth class as \(\$ 150\), write the class limits for all six classes. b. Determine the class boundaries and class widths. c. Find the class midpoints.
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