Question

A consumer organization compared gas mileage figures for several models of cars made in the United States with autos manufactured in other countries. The data are shown in the table: $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Gas Mileage } \\ (\mathrm{m} \mathrm{pg}) \end{array} & \text { Country } & \begin{array}{c} \text { Gas Mileage } \\ (\mathrm{mpg}) \end{array} & \text { Country } \\ \hline 16.9 & \text { U.S. } & 26.8 & \text { U.S. } \\ 15.5 & \text { U.S. } & 33.5 & \text { U.S. } \\ 19.2 & \text { U.S. } & 34.2 & \text { U.S. } \\ 18.5 & \text { U.S. } & 16.2 & \text { Other } \\ 30.0 & \text { U.S. } & 20.3 & \text { Other } \\ 30.9 & \text { U.S. } & 31.5 & \text { Other } \\ 20.6 & \text { U.S. } & 30.5 & \text { Other } \\ 20.8 & \text { U.S. } & 21.5 & \text { Other } \\ 18.6 & \text { U.S. } & 31.9 & \text { Other } \\ 18.1 & \text { U.S. } & 37.3 & \text { Other } \\ 17.0 & \text { U.S. } & 27.5 & \text { Other } \\ 17.6 & \text { U.S. } & 27.2 & \text { Other } \\ 16.5 & \text { U.S. } & 34.1 & \text { Other } \\ 18.2 & \text { U.S. } & 35.1 & \text { Other } \\ 26.5 & \text { U.S. } & 29.5 & \text { Other } \\ 21.9 & \text { U.S. } & 31.8 & \text { Other } \\ 27.4 & \text { U.S. } & 22.0 & \text { Other } \\ 28.4 & \text { U.S. } & 17.0 & \text { Other } \\ 28.8 & \text { U.S. } & 21.6 & \text { Other } \end{array} $$ a) Create graphical displays for these two groups. b) Write a few sentences comparing the distributions.

Short answer

Create histograms or box plots for each group to compare gas mileage. Compare average mileage and variability in the data to see how U.S. cars differ from others.

Step-by-step solution

Separate Data by Country

Start by categorizing the gas mileage data into two groups based on the country of manufacture: U.S. cars and other countries' cars. The U.S. data are: 16.9, 15.5, 19.2, 18.5, 30.0, 30.9, 20.6, 20.8, 18.6, 18.1, 17.0, 17.6, 16.5, 18.2, 26.5, 21.9, 27.4, 28.4, 28.8. The data for cars from other countries are: 26.8, 33.5, 34.2, 16.2, 20.3, 31.5, 30.5, 21.5, 31.9, 37.3, 27.5, 27.2, 34.1, 35.1, 29.5, 31.8, 22.0, 17.0, 21.6.

Choose Appropriate Graphical Displays

Choose graphical displays that would best represent the gas mileage data for comparison. Histograms or box plots are generally suitable for displaying and comparing the distribution of numerical data across groups.

Create Graphical Displays

Create a histogram or box plot for each group (U.S. cars and other countries' cars). Each histogram should display the frequency of cars for various gas mileage intervals. Ensure both histograms or box plots are on the same scale to facilitate easy comparison.

Analyze the Graphical Information

Examine each graphical display you created. Check for the shape of the distribution (e.g., skewness), the central tendency (mean or median), and the spread (range, interquartile range). This will help to compare the gas mileage of cars from the U.S. with those from other countries.

Compare the Distributions

Write a comparative analysis based on your graphical displays. Discuss aspects like the average gas mileage, variability in gas mileage, and any outliers in the data for U.S. auto manufacturers versus other countries. Note if one group generally has higher gas mileage than the other or if there's a wider spread in one group's data.

Key concepts

Graphical Displays in Descriptive Statistics

Graphical displays are powerful tools in statistics that allow us to visually interpret data sets. They provide an immediate sense of the data's distribution and trends, making complex data easier to comprehend. In the context of gas mileage data for cars, graphical displays help us compare and contrast different groups.

When dealing with numerical data like gas mileage, effective graphical tools include histograms and box plots. These tools are instrumental in illustrating data properties such as

• frequency distribution,
• central tendency (mean, median),
• spread of data (range, interquartile range),
• and the presence of outliers.

By organizing data visually, graphical displays facilitate a deeper understanding and provide a foundation for further analysis.

Understanding Histograms as a Graphical Tool

Histograms provide a visual representation of the distribution of numerical data, like gas mileage figures for different cars. Each bar in a histogram represents the frequency, or number of data points, that fall within a particular range of values.

Here is what you can analyze with a histogram:

• Shape: Determine if the distribution is symmetrical, skewed to the left, or skewed to the right based on how the bars are oriented.
• Central Tendency: Identify where most of the data points are clustered along the x-axis.
• Spread: Gauge how wide or narrow the data distribution is, by observing the range and cluster of bars.
• Outliers: Spot any bars that stand significantly apart from others, indicating unusual values.

Understanding these aspects in a histogram allows for a quick comparison between U.S. manufactured cars and other countries' cars in terms of gas mileage.

Box Plots: A Compact Visualization of Data Distribution

Box plots, also known as box-and-whisker plots, offer a concise summary of the data's distribution. They display the dataset's central values and the spread, highlighting any potential outliers.

A box plot consists of a "box" that shows the interquartile range (IQR), which is the spread of the middle 50% of the data.

• Median: The line inside the box represents the median value, providing insight into the data's central tendency.
• Whiskers: These lines extend from the box to the minimum and maximum values, illustrating the data's full range and highlighting any outliers with individual points.
• Comparison: Using box plots for both U.S. and other countries' cars allows for an intuitive comparison of their gas mileage distributions, helping to quickly identify differences in central tendencies, variability, or presence of outliers.

Comparing Distributions with Graphical Displays

When using graphical displays like histograms and box plots, we can not only visualize data but also perform thorough distribution comparisons. This process involves analyzing several aspects.

For the gas mileage data:

• Central Tendency: Determine which group has a higher average or median gas mileage. For instance, by inspecting the median in box plots, you can see which type of cars generally have better fuel efficiency.
• Variability: Assess the spread or dispersion of data. Histograms with longer tails or box plots with a wider range suggest more variability in gas mileage.
• Outliers: Noteworthy outliers can skew perceptions of average fuel efficiency. Identifying them in both graphical tools helps distinguish typical values from anomalies.
• General Trends: A straightforward side-by-side comparison reveals differences in frequency and commonality. It might tell us if cars from other countries tend to have higher gas mileage compared to U.S. cars, based on observed patterns in the data.

By studying these features, we can gain a comprehensive understanding of how different cars perform in terms of gas mileage, helping consumers make informed decisions.