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Chapter 2: Problem 1

For each of the following data sets, create a stem plot and identify any outliers. The miles per gallon rating for 30 cars are shown below (lowest to highest). 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43

Short Answer

Expert verified
Create the stem plot and visually inspect for outliers; none are obvious.

Step by step solution

01

Organize Data into Stems and Leafs

First, we organize the given data into a stem-and-leaf display. The 'stem' represents the tens digit, and the 'leaf' represents the units digit. For example, the number 19 is split into a stem of 1 and a leaf of 9. Write each stem (the tens place of each number), followed by the corresponding leaf (the units place of each number).
02

Create the Stem Plot

Create the stem plot by listing the stems in ascending order as a vertical line. Next to each stem, place the corresponding leaves in ascending order, ensuring each digit aligns with its respective stem. The stem-and-leaf plot for the data should look like this: 1 | 9 9 9 2 | 0 1 1 5 5 5 6 6 8 9 3 | 1 1 2 2 3 4 5 6 7 7 8 8 8 8 4 | 1 3 3
03

Identify the Outliers

Outliers are typically points that deviate significantly from the rest of the data. To identify outliers, one method is to use the 1.5 IQR rule, but a simpler initial visual inspection can help. In this plot, the data is fairly evenly distributed, and no miles per gallon rating stands out as exceptionally higher or lower than the others adjacent to it. There appear to be no clear outliers based on the visual inspection of this stem plot.

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Key Concepts

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

Data Visualization
Understanding data visualization is crucial for analyzing quantitative data efficiently. A stem-and-leaf plot is a robust form of data visualization. It helps in organizing numerical data, giving a quick visual representation of distribution. By splitting each value into a "stem," usually representing the highest place value, and "leaf," representing the next greatest place value, it simplifies complex data sets.
  • The stem-and-leaf plot maintains the original data values, unlike histograms, ensuring every single data point is visible.
  • It provides a clear view of data distribution, clustering, and spread.
  • It's excellent for small to moderate-sized data sets where you need a detailed view without losing any data point.
With practice, you'll quickly spot patterns, notice skewness, and detect gaps or clustering in your data.
Outlier Detection
Detecting outliers is an essential step in data analysis. An outlier is a data point that markedly differs from other observations. In the context of a stem-and-leaf plot, they can be spotted as leaves that seem isolated from other clusters.
  • Visual inspection can reveal these unusual data points, as they will not conform to the general pattern of the dataset.
  • The 1.5 IQR (Interquartile Range) rule is a more quantitative method: calculate the IQR and multiply by 1.5, adding it to the third quartile and subtracting from the first to find potential outliers.
If the outliers are genuine, they can offer insights, like identifying a unique subgroup, errors in data collection, or extraordinary conditions during data capture.
Descriptive Statistics
Descriptive statistics summarize the main features of a dataset. Key metrics include mean, median, mode, range, and standard deviation. When used with stem-and-leaf plots, descriptive statistics help clarify the implications of data points.
  • Mean: The average of the dataset, indicating a central value.
  • Median: The middle number in a sorted data set, effective in understanding the central tendency, especially with skewed data.
  • Mode: The most frequently occurring data point, which the stem-and-leaf plot can highlight clearly.
  • Range: The difference between the highest and lowest values, informing us about the spread of data.
Using these metrics can aid in constructing a comprehensive picture of the data, going beyond mere visualization.
Miles Per Gallon Analysis
Analyzing miles per gallon (MPG) is crucial for understanding vehicle efficiency. MPG metrics are vital for consumers and manufacturers alike to evaluate fuel efficiency, economic decisions, and even environmental impacts.
  • In the provided data set, the MPG ratings range from 19 to 43, hinting at variance in car performance.
  • Higher MPG figures generally suggest better fuel efficiency, aiding eco-friendly vehicle choices.
  • Using a stem-and-leaf plot for MPG aids in quickly comparing car efficiencies and locating clusters of highly efficient vehicles.
Employing these analyses can aid car manufacturers in designing better models and consumers in making informed choices, directly influencing market trends.

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Most popular questions from this chapter

a. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race? b. The 20th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20th percentile in the context of the situation. c. A bicyclist in the 90th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90th percentile in the context of the situation.

The students in Ms. Ramirez鈥檚 math class have birthdays in each of the four seasons. Table 2.40 shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students. $$ \begin{array}{|l|l|}\hline \text { Seasons } & {\text { Number of students }} & {\text { Proportion of population }} \\ \hline \text { Spring } & {8} & {24 \%} \\ \hline \text { Summer } & {9} & {26 \%} \\ \hline \text { Autumn } & {11} & {32 \%} \\ \hline \text { Winter } & {6} & {18 \%} \\\ \hline\end{array} $$

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The mean and median for the data are the same. 3; 4; 5; 5; 6; 6; 6; 6; 7; 7; 7; 7; 7; 7; 7 Is the data perfectly symmetrical? Why or why not?

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system: a. Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an "admissions index" score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12% of high school students in the state. In this context, what percentile does the top 12% represent? b. Students whose GPAs are at or above the 96th percentile of all students at their high school are eligible (called eligible in the local context), even if they are not in the top 12% of all students in the state. What percentage of students from each high school are "eligible in the local context"?
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