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Chapter 4: Problem 38

Create a stem-and-leaf display for these horsepowers of autos reviewed by Consumer Reports one year, and describe the distribution: \(\begin{array}{rrrrr} 155 & 103 & 130 & 80 & 65 \\ 142 & 125 & 129 & 71 & 69 \\ 125 & 115 & 138 & 68 & 78 \\ 150 & 133 & 135 & 90 & 97 \\ 68 & 105 & 88 & 115 & 110 \\ 95 & 85 & 109 & 115 & 71 \\ 97 & 110 & 65 & 90 & \\ 75 & 120 & 80 & 70 & \end{array}\)

Short Answer

Expert verified
The stem-and-leaf display shows a symmetric, multimodal distribution centered around 80-115, with values from 65 to 155 and no significant outliers.

Step by step solution

01

Organize the Data

First, order the data set of car horsepowers in ascending order. This will help make the construction of the stem-and-leaf plot easier. Organize as follows:\[65, 65, 68, 68, 69, 70, 71, 71, 75, 78, 80, 80, 85, 88, 90, 90, 95, 97, 97, 103, 105, 109, 110, 110, 115, 115, 115, 120, 125, 125, 129, 130, 133, 135, 138, 142, 150, 155\]
02

Identify Stems and Leaves

For creating the stem-and-leaf plot, identify the 'stem' as the tens digit and the 'leaf' as the unit digit of each number. In this data set, stems will range from 6 to 15 as the smallest and largest tens values respectively.
03

Construct the Stem-and-Leaf Display

Draw the stem-and-leaf display. List stems (tens digit) in a column, and next to each stem, write the leaves (units digit). Fill in the values based on the ordered list:\[\begin{array}{cl}6: & 5, 5, 8, 8, 9 \7: & 0, 1, 1, 5, 8 \8: & 0, 0, 5, 8 \9: & 0, 0, 5, 7, 7 \10: & 3, 5, 9 \11: & 0, 0, 5, 5, 5 \12: & 0, 5, 5, 9 \13: & 0, 3, 5, 8 \14: & 2 \15: & 0, 5 \\end{array}\]
04

Describe the Distribution

Examine the stem-and-leaf plot and describe the distribution of horsepowers. Observed characteristics include: - **Shape:** The distribution is approximately symmetric with several peaks suggesting a multimodal distribution. - **Center:** Most values appear grouped around the 80-115 range. - **Spread:** Horsepower values range from 65 to 155. - **Outliers/Unusual Features:** No significant outliers or unusual concentrations of leaves in a particular stem.

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Key Concepts

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

Data Visualization
Data visualization is a powerful method for understanding complex datasets. One of the simplest and most effective forms of data visualization is the stem-and-leaf plot. This tool helps us see the shape and distribution of the data instantly.

The strength of a stem-and-leaf plot lies in its ability to organize large amounts of data. It presents the data in a way that allows us to see individual values while also understanding the overall distribution. It's like a histogram, but more detailed. You can see each occurrence of each value in the dataset.

When creating a stem-and-leaf plot, the 'stem' represents the leading digits and the 'leaf' represents the final digit of each number. This format is especially useful with numerical data, such as the horsepower of cars as seen in our exercise.
  • Helps in quickly identifying the mode, median, and potential outliers
  • Simplifies large sets of data into a visual layout
  • Highlights how data points cluster around certain values
Distribution Analysis
Distribution analysis involves understanding and interpreting the spread and shape of data. Using tools like the stem-and-leaf plot, we can perform a detailed distribution analysis.

In this exercise, we examine the data鈥檚 distribution to identify features such as shape, central tendency, and variability.

A key aspect to note is the shape of the distribution. In this case, the distribution was approximately symmetric with peaks, indicating it may be multimodal. This shows that several horsepower values occur more frequently, creating multiple peaks in our data.
  • Shape: Symmetric distributions indicate balance, while skewed distributions suggest data leans more towards one side.
  • Center: The median helps us understand where most of our data falls. Here, it's around the 80-115 range.
  • Spread: The range reveals the gap between the minimum and maximum values, such as 65 to 155 horsepower.
Understanding how data points are distributed helps us predict patterns and behaviors in the dataset.
Univariate Data
Univariate data refers to a dataset consisting of observations on a single variable. The focus in univariate data is to describe, analyze, and summarize the set of data points.

In our example with horsepower data, we're dealing with univariate data because all data points are measurements of the same attribute: horsepower of cars.

Examining univariate data involves looking at individual data points as well as the entire distribution. This includes calculating measures of central tendency like mean, median, and mode, and exploring the spread of the data through range and interquartile range.
  • It鈥檚 crucial to visualize univariate data to uncover underlying patterns and features.
  • Helps in identifying the most frequent occurrences and potential anomalies or outliers.
  • Serves as a foundation for more complex multivariate analysis.
By thoroughly analyzing univariate data, we gain insights that guide further inquiries and investigations.

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Most popular questions from this chapter

A report from the U.S. Department of Justice (www.ojp.usdoj.gov/bjs/) reported the percent changes in federal prison populations in 21 northeastern and midwestern states during 2005. Using appropriate graphical displays and summary statistics, write a report on the changes in prison populations. \(\begin{array}{l|c|l|c} \text { State } & \begin{array}{l} \text { Percent } \\ \text { Change } \end{array} & \text { State } & \begin{array}{c} \text { Percent } \\ \text { Change } \end{array} \\ \hline \text { Connecticut } & -0.3 & \text { Iowa } & 2.5 \\ \text { Maine } & 0.0 & \text { Kansas } & 1.1 \\ \text { Massachusetts } & 5.5 & \text { Michigan } & 1.4 \\ \text { New Hampshire } & 3.3 & \text { Minnesota } & 6.0 \\ \text { New Jersey } & 2.2 & \text { Missouri } & -0.8 \\ \text { New York } & -1.6 & \text { Nebraska } & 7.9 \\ \text { Pennsylvania } & 3.5 & \text { North Dakota } & 4.4 \\ \text { Rhode Island } & 6.5 & \text { Ohio } & 2.3 \\ \text { Vermont } & 5.6 & \text { South Dakota } & 11.9 \\ \text { Illinois } & 2.0 & \text { Wisconsin } & -1.0 \\ \text { Indiana } & 1.9 & & \\ \hline \end{array}\)

For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand. $$ \begin{array}{ll} {\text { Set 1 }} & {\text { Set 2 }} \\ \hline \text { a) } 4,7,7,7,10 & 4,6,7,8,10 \\ \text { b) } 100,140,150,160,200 & 10,50,60,70,110 \\ \text { c) } 10,16,18,20,22,28 & 48,56,58,60,62,70 \end{array} $$

During his 20 seasons in the NHL, Wayne Gretzky scored \(50 \%\) more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: \(\begin{aligned} &79,80,80,80,74,80,80,79,64,78,73,78,74,45,81,48,80, \\ &82,82,70 \end{aligned}\) a) Create a stem-and-leaf display for these data, using split stems. b) Describe the shape of the distribution. c) Describe the center and spread of this distribution. d) What unusual feature do you see? What might explain this?

Exercise 22 looked at distances PGA golfers can hit the ball. The standard deviation of these average drive distances is \(9.3\) yards, and the quartiles are \(Q_{1}=282\) yards and \(Q_{3}=294\) yards. a) Write a sentence or two describing the spread in distances based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

The frequency table shows the heights (in inches) of 130 members of a choir. \(\begin{array}{c|c|c|c} \text { Height } & \text { Count } & \text { Height } & \text { Count } \\ \hline 60 & 2 & 69 & 5 \\ 61 & 6 & 70 & 11 \\ 62 & 9 & 71 & 8 \\ 63 & 7 & 72 & 9 \\ 64 & 5 & 73 & 4 \\ 65 & 20 & 74 & 2 \\ 66 & 18 & 75 & 4 \\ 67 & 7 & 76 & 1 \\ 68 & 12 & & \end{array}\) a) Find the median and IQR. b) Find the mean and standard deviation. c) Display these data with a histogram. d) Write a few sentences describing the distribution.
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