Question

In March 2006, 16 gas stations in Grand Junction, CO, posted these prices for a gallon of regular gasoline: \(\begin{array}{llll} 2.22 & 2.21 & 2.45 & 2.24 \\ 2.27 & 2.28 & 2.27 & 2.23 \\ 2.26 & 2.46 & 2.29 & 2.32 \\ 2.36 & 2.38 & 2.33 & 2.27 \end{array}\) a) Make a stem-and-leaf display of these gas prices. Use split stems; for example, use two \(2.2\) stems-one for prices between \(\$ 2.20\) and \(\$ 2.24\) and the other for prices from \(\$ 2.25\) to \(\$ 2.29 .\) b) Describe the shape, center, and spread of this distribution. c) What unusual feature do you see?

Short answer

The stem-and-leaf display shows a bimodal distribution with frequent occurrence at \$2.27, an unusual feature.

Step-by-step solution

Organize Prices in Order

To create a stem-and-leaf plot, we first organize the gas prices in ascending order: \[2.21, 2.22, 2.23, 2.24, 2.26, 2.27, 2.27, 2.27, 2.28, 2.29, 2.32, 2.33, 2.36, 2.38, 2.45, 2.46\]

Create the Stems

Since we have prices mainly in the range of \\(2.20 to \\)2.46, we choose stems based on the tens and ones digits, deciding to split the \(2.2\) stem as follows:- Stem \(2.2\) for prices \( ext{\\(2.20} - ext{\\)2.24}\)- Stem \(2.2\) for prices \( ext{\\(2.25} - ext{\\)2.29}\)- Stem \(2.3\) for prices \( ext{\\(2.30} - ext{\\)2.39}\)- Stem \(2.4\) for prices \( ext{\\(2.40} - ext{\\)2.49}\)

Assemble the Leaf Components

For each price, the last digit is a leaf attached to its corresponding stem:- Stem \(2.2\) from \(2.20\) to \(2.24\): Leaves \(1, 2, 3, 4\)- Stem \(2.2\) from \(2.25\) to \(2.29\): Leaves \(6, 7, 7, 7, 8, 9\)- Stem \(2.3\): Leaves \(2, 3, 6, 8\)- Stem \(2.4\): Leaves \(5, 6\)

Construct the Stem-and-Leaf Display

Now, we can display the organized prices in a stem-and-leaf plot: 2.2 | 1 2 3 4 2.2 | 6 7 7 7 8 9 2.3 | 2 3 6 8 2.4 | 5 6

Describe Shape, Center, and Spread

The shape of the distribution is approximately bimodal with peaks around \(2.27\) and \(2.30\). The center can be estimated at around \(2.27\), and the spread ranges from \(2.21\) to \(2.46\). This suggests a moderate distribution concentrated in two groups.

Analyze Unusual Features

The unusual feature observed is that the value \(2.27\) appears frequently—three times, suggesting a common price point among the gas stations.

Key concepts

Data Visualization

Data visualization is a critical tool in understanding numerical data. A common method for visualizing data like gas prices is the stem-and-leaf plot. This type of plot is particularly useful as it maintains the original data values while organizing them in a way that makes patterns clear.

In the exercise, a stem-and-leaf plot is constructed by first ordering gas prices from the lowest to the highest. Then, we create 'stems,' which are the leading digits of the data points. The 'leaves' follow as the last digit of each data point. For instance, the price \(2.21\) has a stem of \(2.2\) and a leaf of \(1\). Splitting the stems—such as \(2.2\)—into smaller ranges like \(2.20 - 2.24\) and \(2.25 - 2.29\) helps in spotting trends within tightly packed data.

By visualizing the prices in this format, we quickly identify clusters (or modes) and outliers. This visual method simplifies understanding the general structure and distribution of the gas prices, making it easier to derive insights.

Distribution Analysis

Distribution analysis is the process of examining data to understand its underlying structure. When analyzing a stem-and-leaf plot, we investigate the distribution of gas prices to comprehend their arrangement, peaks, and spread.

A key observation from the provided stem-and-leaf plot is its bimodal shape. This means there are two distinct peaks, which in this case are centered around the prices \(2.27\) and \(2.30\). The bimodal distribution hints that prices tend to cluster around these points, rather than being uniformly distributed or centered at a single point.

Understanding the distribution helps in identifying where most data points lie, and in this case, identifying potential common pricing strategies gas stations might follow. Additionally, it highlights the presence of an unusual feature—multiple appearances of \(2.27\)—which indicates a shared pricing strategy at certain stations.

Descriptive Statistics

Descriptive statistics provide a summary of the distribution of data, including measures like the center, spread, and any prominent features or anomalies.

For the gas price data, descriptive statistics involve estimating the central tendency—often represented by the median or mean. Here, the center of the distribution is estimated around \(2.27\), a frequent price point. This gives an idea of a typical value where most prices converge.

The spread measures the range from the lowest price \(2.21\) to the highest price \(2.46\). The moderate spread suggests that prices did not vary widely, indicating stability in pricing within this set of data. Additionally, the repeated occurrence of \(2.27\) shows a notable pattern that highlights commonality in pricing strategies among some stations, an anomaly that could have economic implications.

By summarizing the data's description through these basic statistics, one gains a clearer and more accessible understanding of the entire distribution, making data interpretation straightforward and manageable.