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Chapter 1: Problem 6

Here are the exam scores of 10 students in a statistics class: $$ \begin{array}{llllllllll} 50 & 35 & 41 & 97 & 76 & 69 & 94 & 91 & 23 & 65 \end{array} $$ To make a stemplot of these data, you would use stems a. \(2,3,4,5,6,7,9\). b. \(2,3,4,5,6,7,8,9\). c. \(20,30,40,50,60,70,80,90\).

Short Answer

Expert verified
The correct answer is a. \(2,3,4,5,6,7,9\).

Step by step solution

01

Understanding Stemplot Basics

A stemplot, or stem-and-leaf plot, is a method used to display quantitative data where the stem represents the leading digits and the leaf represents the trailing digit. For example, in the number 41, '4' would be the stem and '1' would be the leaf.
02

Identifying Possible Stems

Let's examine the given data set: 50, 35, 41, 97, 76, 69, 94, 91, 23, 65. We note the tens digit in each score to determine the stems. These tens digits are 5, 3, 4, 9, 7, 6, 9, 9, 2, and 6, respectively.
03

Listing Unique Stems

Look for the distinct tens digits from the exam scores to use as stems. The unique tens digits from the collection 5, 3, 4, 9, 7, 6, 9, 9, 2, and 6 are 2, 3, 4, 5, 6, 7, and 9.
04

Selecting Correct Answer

Compare the list of unique stems to the choices provided: (a) \(2,3,4,5,6,7,9\). (b) \(2,3,4,5,6,7,8,9\). (c) \(20,30,40,50,60,70,80,90\). The correct set of stems, based on our unique tens digits, is \(2,3,4,5,6,7,9\), which matches option (a).

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Key Concepts

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

Quantitative Data Representation
Quantitative data representation involves executing methods to visualize numerical data effectively. When you represent data quantitatively, you can gain insights and identify patterns, trends, or outliers within a dataset.
This presentation of data can come in various forms:
  • Dot Plots: Use dots to represent individual data points on a number line.
  • Histograms: Organize data into continuous intervals or bins, displaying frequency using bars.
  • Box Plots: Highlight the distribution of data based on a five-number summary - minimum, first quartile, median, third quartile, and maximum.
  • Stem-and-Leaf Plots: Relate to our main topic, where data is split into 'stem' and 'leaf', showing frequency distribution.
These representations make it easier to interpret complex data by providing visual aids that can summarize a large collection of numbers. This approach makes significant trends and patterns visible at a glance.
Stem-and-Leaf Plot
A stem-and-leaf plot, also known as a stemplot, is a simple yet powerful tool used in statistics to display quantitative data. It organizes the data in such a way that the values themselves tell a story.
Here's how it works:
  • The 'stem' represents all but the last digit. For instance, for the score 97, the stem would be 9.
  • The 'leaf' represents the last digit of the number. For the same score 97, the leaf would be 7.
When creating a stem-and-leaf plot:
  • Arrange data from least to greatest for consistency.
  • List stems in a vertical column, even if they have no corresponding leaves.
  • Align leaves to the right of their respective stems, ensuring clarity.
This method is particularly useful for small to medium-sized datasets and provides an easy way to see the shape of data distribution while maintaining the original data values.
Data Visualization in Statistics
Data visualization in statistics is the science of transforming numbers into a visual context, like a graph or chart, to make the data easier to understand.
This is crucial because:
  • It uncovers trends and patterns not apparent in purely numerical form.
  • Enables quick and efficient analysis of large datasets.
  • Facilitates comparisons across different datasets or variable categories.
Visual tools such as histograms, box plots, line graphs, and stem-and-leaf plots are commonly used to depict statistical information. By presenting data visually, it becomes much more interactive and comprehensible, allowing users to draw informed conclusions quickly without sifting through numbers.
Statistics is about providing data clarity, and visualization is a key component in turning raw data into meaningful insights.

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Most popular questions from this chapter

Nint endo and Laparoscopic Skills. In laparoscopic surgery, a video camera and several thin instruments are inserted into the patient's abdominal cavity. The surgeon uses the image from the video camera positioned inside the patient's body to perform the procedure by manipulating the instruments that have been inserted. It has been found that the Nintendo Wii \({ }^{\text {meproduces }}\) the movements required in laparoscopic surgery more closely than other video games with its motion-sensing interface. If training with a Nintendo Wiim can improve laparoscopic skills, it can complement the more expensive training on a laparoscopic simulator. Forty-two medical residents were chosen, and all were tested on a set of basic laparoscopic skills. Twenty-one were selected at random to undergo systematic Nintendo Wii \({ }^{\text {nu }}\) training for one hour a day, five days a week, for four weeks. The remaining 21 residents were given no Nintendo Wiim training and asked to refrain from video games during this period. At the end of four weeks, all 42 residents were tested again on the same set of laparoscopic skills. One of the skills involved a virtual gall bladder removal, with several performance measures including time to complete the task recorded. Here are the improvement (before-after) times in seconds after four weeks for the two groups: a. In the context of this study, what do the negative values in the data set mean? b. Back-to-back stemplots can be used to compare the two samples. That is, use one set of stems with two sets of leaves, one to the right and one to the left of the stems. (Draw a line on either side of the stems to separate stems and leaves.) Order both sets of leaves from smallest at the stem to largest away from the stem. Complete the back-to-back stemplot started below. The data have been rounded to the nearest 10 , with stems being 100 s and leaves being 10 s. The stems have been split. The first control observation corresponds to \(-80\) and the next two to \(-30\) and \(-10 .\) c. Report the approximate midpoints of both groups. Does it appear that the treatment has resulted in a greater improvement in times than seen in the control group? (To better understand the magnitude of the improvements, note that the median time to complete the task on the first occasion was 11 minutes and 40 seconds, using the times of all 42 residents.)

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