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Chapter 2: Problem 48

The following data give the time (in minutes) that each of 20 students waited in line at their bookstore to pay for their textbooks in the beginning of Spring 2012 semester. (Note: To prepare a stem-andleaf display, each number in this data set can be written as a two-digit number. For example, 8 can be written as 08 , for which the stem is 0 and the leaf is \(8 .\) ) \(\begin{array}{rrrrrrrrrr}15 & 8 & 23 & 21 & 5 & 17 & 31 & 22 & 34 & 6 \\ 5 & 10 & 14 & 17 & 16 & 25 & 30 & 3 & 31 & 19\end{array}\) Construct a stem-and-leaf display for these data. Arrange the leaves for each stem in increasing order.

Short Answer

Expert verified
The stem-and-leaf display for this data set is: \n 0 | 35568 \n 1 | 0456779 \n 2 | 1235 \n 3 | 0144

Step by step solution

01

Order the Data

Firstly, order the given data in ascending order. The sorted dataset is: 3, 5, 5, 6, 8, 10, 14, 15, 16, 17, 17, 19, 21, 22, 23, 25, 30, 31, 31, 34.
02

Identify the Stem and Leaf

In a stem-and-leaf plot, the stem represents the highest value place (for example tens) and the leaf represents the lowest place value (for example ones). In this case, the tens place will be the stem and the ones place will be the leaf.
03

Construct the Stem-and-Leaf Display

Group the dataset by tens place and then in each group, write down the leafs. For this data set, the stem-and-leaf display would be as follows: \n 0 | 35568 \n 1 | 0456779 \n 2 | 1235 \n 3 | 0144

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Key Concepts

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

Data Visualization
Stem-and-leaf plots are a simple and effective way to visualize numerical data. By organizing the data in this format, it presents a clear picture of distribution. Stem-and-leaf plots display quantitative data in a way that helps you easily see the shape of the data set. The benefits include:
  • Quickly identifying the minimum and maximum values.
  • Spotting trends or patterns, such as clustering of values.
  • Easily recognizing any outliers.
Data visualization techniques like stem-and-leaf plots are foundational tools in statistics. They transform raw numbers into visual insights, making it easier for you to interpret the data effectively.
Descriptive Statistics
Descriptive statistics summarize or describe characteristics of a data set. With stem-and-leaf plots, you can quickly understand key details of your data. For example, in the given exercise, the data set summarized students' wait times. You can gather essential information about how wait times are spread across different intervals.
Stem-and-leaf plots specifically help identify descriptive statistics such as:
  • Central tendency: Recognizing where most data points cluster.
  • Spread: Observing the range of wait times.
  • Frequency: Seeing how often specific wait times occur.
Understanding these aspects of your data provides valuable insights into overall patterns and individual data points.
Numerical Data Analysis
Numerical data analysis refers to techniques used to interpret data points to derive meaningful insights. Stem-and-leaf plots play a pivotal role in making numerical data analysis easier for observational and statistical purposes. By organizing the numbers into easily readable sections, you can highlight important features of the data, such as variability and distribution.
In the context of the exercise, analyzing wait-times helps in:
  • Reducing complex data into a manageable format.
  • Identifying the symmetry or skewness of the distribution.
  • Facilitating comparison with other similar data sets.
This simplification is crucial for quickly drawing conclusions and informing decisions based on the data presented.

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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 \(7-9\).

The following data give the results of a sample survey. The letters \(\mathrm{Y}, \mathrm{N}\), and \(\mathrm{D}\) represent the three categories. \(\begin{array}{llllllllll}\mathrm{D} & \mathrm{N} & \mathrm{N} & \mathrm{Y} & \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{Y} & \mathrm{D} & \mathrm{Y} \\\ \mathrm{Y} & \mathrm{Y} & \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{N} & \mathrm{Y} \\ \mathrm{N} & \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{D} & \mathrm{N} & \mathrm{Y} & \mathrm{Y} & \mathrm{Y} & \mathrm{Y} \\ \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{N} & \mathrm{Y} & \mathrm{Y} & \mathrm{N} & \mathrm{N} & \mathrm{D} & \mathrm{Y}\end{array}\) C. a. Prepare a frequency distribution table. c. What percentage of the elements in this sample belong to category Y? d. What percentage of the elements in this sample belong to category \(\mathrm{N}\) or \(\mathrm{D}\) ? e. Draw a pie chart for the percentage distribution. b. Calculate the relative frequencies and percentages for all categories.

Why do we need to group data in the form of a frequency table? Explain briefly.

Following are the total yards gained rushing during the 2012 season by 14 running backs of 14 college football teams. \(\begin{array}{rrrrrrr}745 & 921 & 1133 & 1024 & 848 & 775 & 800 \\ 1009 & 1275 & 857 & 933 & 1145 & 967 & 995\end{array}\) Prepare a stem-and-leaf display. Arrange the leaves for each stem in increasing order.

The following data give the number of text messages sent on 40 randomly selected days during 2012 by a high school student: \(\begin{array}{llllllllll}32 & 33 & 33 & 34 & 35 & 36 & 37 & 37 & 37 & 37 \\\ 38 & 39 & 40 & 41 & 41 & 42 & 42 & 42 & 43 & 44 \\ 44 & 45 & 45 & 45 & 47 & 47 & 47 & 47 & 47 & 48 \\ 48 & 49 & 50 & 50 & 51 & 52 & 53 & 54 & 59 & 61\end{array}\) a. Construct a frequency distribution table. Take 32 as the lower limit of the first class and 6 as the class width. b. Calculate the relative frequency and percentage for each class. c. Construct a histogram for the frequency distribution of part a. d. On what percentage of these 40 days did this student send more than 44 text messages?
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