91Ó°ÊÓ

Do running times of American movies differ somehow from times of French movies? The authors investigated this question by randomly selecting 25 recent movies of each type, resulting in the following running times: \(\begin{array}{lrrrrrrr}\text { Am: } & 94 & 90 & 95 & 93 & 128 & 95 & 125 \\\ & 91 & 104 & 116 & 162 & 102 & 90 & 110 \\ & 92 & 113 & 116 & 90 & 97 & 103 & 95 \\ & 120 & 109 & 91 & 138 & & & \\ \text { Fr: } & 123 & 116 & 90 & 158 & 122 & 119 & 125 \\ & 90 & 96 & 94 & 137 & 102 & 105 & 106 \\ & 95 & 125 & 122 & 103 & 96 & 111 & 81 \\ & 113 & 128 & 93 & 92 & & & \end{array}\) Construct a comparative stem-and-leaf display by listing stems in the middle of your paper and then placing the Am leaves out to the left and the Fr leaves out to the right. Then comment on interesting features of the display.

Step by step solution

01

Create the Stems

To create a stem-and-leaf plot, first determine the stems from both datasets. Stems are typically the leading digits of the numbers. Observing the movie durations, the stem will mostly be the first two digits, ranging from 8 to 16.

02

Add American Movie Leaves

Next, list the stems vertically. For each running time from the American movies, write the trailing digit (leaf) to the left of the corresponding stem. For example, for 94, write '4' next to the stem '9'.

03

Add French Movie Leaves

Similarly, for each running time from French movies, write the trailing digit (leaf) to the right of the corresponding stem. For example, for 123, write '3' next to the stem '12'.

04

Align and Check the Leaves

Make sure all leaves for both American and French movies are aligned with the correct stem. Double-check that each running time is appropriately represented to avoid missing any data.

05

Analyze the Display

Examine the completed stem-and-leaf plot. Look for patterns such as clustering, gaps, or differences in distributions between American and French movie times. Discuss any apparent differences or similarities.

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Key Concepts

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

Understanding Data Visualization

A stem-and-leaf plot is a method used in data visualization. It helps organize and display data so that underlying patterns and structures can easily be seen. In the context of comparing movie running times, this type of plot provides a clear snapshot of each dataset's distribution.
The stem represents the main part of each number (usually the leading digits). In our exercise, these correspond to tens and hundreds in movie running times.
The leaf is the trailing digit of a number. By aligning these components, you create a map that shows the frequency and spread of data points.
Stem-and-leaf plots are especially useful because they retain the original data values while showing data in a way that is easy to understand. This means you can see both the frequency and actual data points without losing any detail.

Performing Comparative Analysis

Comparative analysis using stem-and-leaf plots involves placing two datasets side by side to identify differences or similarities. In our exercise, you compare American and French movie running times by plotting them in opposite directions relative to their shared stems.
This side-by-side arrangement presents a clear, comparative view, allowing you to see how data from each group compares near each stem.
Key observations might include:

• Clustering, which indicates that a lot of data points are close together.
• Gaps that signify ranges with few or no data points.
• Variations in spread or distribution, showing whether one group tends to have longer or shorter running times.

By analyzing these patterns, you draw conclusions regarding possible differences in movie lengths between the two countries.

Exploring Descriptive Statistics

Descriptive statistics offer a way to summarize and describe essential features of a dataset using key figures. When combined with stem-and-leaf plots, they provide a comprehensive understanding of data.
Common descriptive measures include:

• The mean, or average, which provides a central point of the data.
• The median, which divides the dataset into two equal parts.
• The mode, or the most frequently occurring value.
• Range, showing the spread or variability in the dataset.

Looking at the exercise, descriptive statistics can be calculated for each movie category. By doing so, you obtain a mathematical foundation for differences observed visually in the stem-and-leaf plot.
For instance, if the average running time is higher for French movies, this might explain clustering patterns observed in their stems. These statistics allow students to make informed interpretations backed by quantifiable measurements.