Question

Accidents involving drunk drivers account for about \(40 \%\) of all deaths on the nation's highways. The table tracks the number of alcohol-related fatalities for 24 years. (www.madd.org) $$ \begin{array}{c|c|c|c} \text { Year } & \text { Deaths (thousands) } & \text { Year } & \text { Deaths (thousands) } \\ \hline \mathbf{1 9 8 2} & 26.2 & \mathbf{1 9 9 4} & 17.3 \\ \mathbf{1 9 8 3} & 24.6 & \mathbf{1 9 9 5} & 17.7 \\ \mathbf{1 9 8 4} & 24.8 & \mathbf{1 9 9 6} & 17.7 \\ \mathbf{1 9 8 5} & 23.2 & \mathbf{1 9 9 7} & 16.7 \\ \mathbf{1 9 8 6} & 25.0 & \mathbf{1 9 9 8} & 16.7 \\ \mathbf{1 9 8 7} & 24.1 & \mathbf{1 9 9 9} & 16.6 \\ \mathbf{1 9 8 8} & 23.8 & \mathbf{2 0 0 0} & 17.4 \\ \mathbf{1 9 8 9} & 22.4 & \mathbf{2 0 0 1} & 17.4 \\ \mathbf{1 9 9 0} & 22.6 & \mathbf{2 0 0 2} & 17.5 \\ \mathbf{1 9 9 1} & 20.2 & \mathbf{2 0 0 3} & 17.1 \\ \mathbf{1 9 9 2} & 18.3 & \mathbf{2 0 0 4} & 16.9 \\ \mathbf{1 9 9 3} & 17.9 & \mathbf{2 0 0 5} & 16.9 \end{array} $$ a) Create a stem-and-leaf display or a histogram of these data. b) Create a timeplot. c) Using features apparent in the stem-and-leaf display (or histogram) and the timeplot, write a few sentences about deaths caused by drunk driving.

Short answer

Deaths decreased over time, showing improvements and awareness in road safety.

Step-by-step solution

Understand the Problem

We need to create two types of visual representations (a stem-and-leaf display or histogram, and a timeplot) and interpret them for the given data about alcohol-related fatalities over a span of 24 years.

Organize Data for Stem-and-Leaf Display

A stem-and-leaf display requires us to split each data point into two parts: the 'stem' (usually the leading digit or digits) and the 'leaf' (usually the last digit). Here, 'stems' can be the tens digit and 'leaves' the units digit of the death figures.

Create the Stem-and-Leaf Display

From the given data, our stems will be 16, 17, 18, 19, 20, 22, 23, 24, 25, and 26. Each leaf is the unit's position of each death count in thousands. For example, the number 26.2 will have a stem of 26 and a leaf of 2.

Organize Data for Histogram

Histograms involve dividing the data into intervals (bins) and counting how many data points fall into each interval. Choose appropriate bin widths (e.g., 1,000) for the given range (16.6 to 26.2 thousands).

Create the Histogram

Count the number of deaths in each bin over the range (16-27 thousands) and draw bars showing the frequency of deaths per bin. For example, if four data points fall between 16.6 and 17.5, the height of that bar is 4.

Create the Timeplot

Plot the number of deaths on the y-axis against the corresponding years on the x-axis. Each year becomes a point on the timeplot. Connect the points linearly to depict trends over time.

Analyze the Stem-and-Leaf Display and Histogram

The stem-and-leaf display or histogram shows the distribution of the fatality numbers. Analyze whether there are apparent clusters, gaps, or outliers that indicate trends or shifts over years.

Analyze the Timeplot

The timeplot provides a chronological view of data. Look for overall trends, such as increases or decreases in alcohol-related deaths over time. Observations can include periods of stability or fluctuations in fatalities.

Conclusion from Analyses

The stem-and-leaf display and histogram may reveal that most death counts are concentrated between the late 16,000s and early 18,000s. The timeplot shows a general decrease over the years with some fluctuations. This indicates a reduction in fatalities due to increasing awareness and preventive measures over time.

Key concepts

Stem-and-Leaf Display

A stem-and-leaf display is a great way to see patterns in numerical data. Imagine dividing a number into two parts: the stem and the leaf. For example, take a number like 24.6. In this display, 24 is the stem, and 6 is the leaf. This separates the tens and ones digit in terms of thousands of deaths, making it easier to analyze and compare data quickly.

To create one, you first list the stems in a vertical column, then attach each leaf to its respective stem. For the dataset involving drunk driving fatalities, stems could be numbers like 16, 17, etc., representing 16,000 deaths and higher.

• This display helps spot clusters, highlighting where data points are closely packed.
• It can reveal gaps indicating where data points are sparse.
• It allows the identification of outliers or unusual data points that deviate significantly.

It’s like a simple histogram, but it retains the actual data, which is helpful when you need precise details.

Histogram

Histograms serve a similar purpose to stem-and-leaf displays by showcasing data distribution, but they use bars to represent frequency. Imagine having a bar graph where each bar covers a range of data, known as bins. For instance, in this scenario, a bin could cover the range from 16.6 to 17.5 thousand deaths. The height of each bar indicates how many data points fall within that range.

To create a histogram:

• First, decide on the bin width. For our data example, a bin width of 1,000 might suit the range from 16,000 to 26,000.
• Count how many entries fall into each bin. If several years had fatalities between 17,000 and 18,000, a bar represents that frequency.

This visualization is effective because:

• It helps to see the overall shape of the data distribution – whether it's skewed left or right, uniform, or normal.
• Unlike simple bar charts, the bins in histograms touch each other, emphasizing the continuous nature of the data.

Overall, histograms summarize data visually, allowing for quick comprehension of large datasets.

Time Series Analysis

Time series analysis focuses on reviewing and analyzing data points collected or recorded over time. In the context of the 24-year data trend on alcohol-related fatalities, a timeplot is a key tool. Imagine plotting the number of fatalities each year on a graph, with years on the x-axis and the number of deaths on the y-axis.

Why is a timeplot useful?

• It shows the data trend over time, making it easier to spot patterns, such as decreases or increases in fatalities.
• It highlights seasons and cycles within the data. For example, periods where fatalities dropped significantly or rose sharply can be identified.
• Trends like overall reduction or increase give insight into long-term changes.

After plotting each point by year and connecting them, you can observe the overall direction of the trend. In our dataset of alcohol-related fatalities, a general decline observed indicates positive developments over time, likely attributable to increased prevention measures and awareness.