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Chapter 1: Problem 6

Do running times of American movies differ somehow from running times of French movies? The author investigated this question by randomly selecting 25 recent movies of each type, resulting in the following running times: \(\begin{array}{lrrrrrrrrrrrrr}\text { Am: } & 94 & 90 & 95 & 93 & 128 & 95 & 125 & 91 & 104 & 116 & 162 & 102 & 90 \\ & 110 & 92 & 113 & 116 & 90 & 97 & 103 & 95 & 120 & 109 & 91 & 138 & \\ \text { Fr: } & 123 & 116 & 90 & 158 & 122 & 119 & 125 & 90 & 96 & 94 & 137 & 102 & \\ & 105 & 106 & 95 & 125 & 122 & 103 & 96 & 111 & 81 & 113 & 128 & 93 & 92\end{array}\) 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.

Short Answer

Expert verified
French movies show a tendency for longer durations while American movies are more balanced across the timeline.

Step by step solution

01

Prepare the Data

First, organize the running times by listing each set of movie running times separately. For American (Am) movies: 94, 90, 95, 93, ..., 138. For French (Fr) movies: 123, 116, 90, 158, ..., 92. Make sure all data entries are accounted for as they will be used to create the stem-and-leaf plot.
02

Identify the Stems

The running times for both American and French movies range from 81 to 162. The stem will be the tens digit for the whole data set, resulting in stems: 8, 9, 10, 11, 12, 13, 14, 15, and 16. This will be the central part of the stem-and-leaf plot.
03

Create Leaves for American Movies

For each stem, list the corresponding ones digits of American movie running times to the left. For example, for stem 9, American data has leaves: 4, 0, 5, 3, 0, 2. Always sort the leaves in ascending order: \( 0, 0, 2, 3, 4, 5 \). Repeat this for all stems.
04

Create Leaves for French Movies

For each stem, list the corresponding ones digits of French movie running times to the right. For example, for stem 9, French data has leaves: 0, 6, 4, 5, 6, 2, 3. Sort the leaves into ascending order: \( 0, 2, 3, 4, 5, 6, 6 \). Repeat this for all stems.
05

Draw the Stem-and-Leaf Plot

Draw a vertical line down the middle of your page. List all of the stems (8 to 16) vertically along this line. To the left of each stem, write the leaves for American movies, and to the right, write the leaves for French movies. Ensure all leaves are ordered in ascending fashion.
06

Analyze the Display

Examine the plot for interesting features, such as cluster regions, spread, or outliers. For example, French movies tend to have many entries above 12, while American movies have more balance across the stems. Outliers if any, can provide insights into differences between the two data sets.

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Key Concepts

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

Comparative Analysis
Comparative analysis is a method used to identify the differences and similarities between two or more sets of data. In this context, we are comparing the running times of American and French movies. By constructing a stem-and-leaf plot, you can visually assess and compare the distributions of these two datasets. This helps to identify patterns such as commonalities, differences, and possible trends.

In our movie running time data, the analysis is facilitated by using a common stem because it allows for direct comparison.
  • The American movies tend to have running times more evenly spread across different stems, indicating a more varied set of running times.

  • French movies appear to have more running times clustered in the higher ranges (stems 12, 13, 15), suggesting a tendency for longer films.

By examining the comparative features and the clustering tendencies, such analyses can lend insights into cultural or industry differences affecting movie durations.
Data Visualization
Data visualization is crucial in interpreting and comparing datasets effectively. A stem-and-leaf plot offers a simple yet efficient means of visually representing numerical data in a compact form. Unlike other plots, it retains the original data values, allowing for precise comparisons and analyses.

To make a stem-and-leaf plot:
  • Write the stems, which are derived from the tens digits of the movie running times, vertically down the center of the plot.

  • To the left of each stem, list the ones digits for American movies. Do the same for French movies on the right side.

  • Ensure the leaves are sorted in ascending order, enabling easy identification of data distributions.

Through this simple structure, you can immediately spot trends in data, such as clusters, gaps, or possible anomalies. For example, if the leaves for French movies are noticeably greater than those for American, you can infer potential differences in running time tendencies, as supported by the data.
Statistical Distributions
Understanding statistical distributions is key in analyzing datasets like movie running times. Distributions help to inform us about how data values are spread and what patterns are evident across a dataset.

A stem-and-leaf plot gives a clear visual representation of these distributions. It shows:
  • How frequently certain running times occur by looking at the density of the leaves for each stem.

  • Potential central tendencies, such as the median or the most common data points, which can appear as clusters.

  • The spread of the data, indicating diversity in movie lengths.

By analyzing the statistical distributions displayed by the stem-and-leaf plot, one might notice French movies demonstrating a trend toward longer durations. Meanwhile, American films show a wider spread and less concentration toward higher running times, hinting at possibly shorter typical durations or more variety in their lengths. Such insights are valuable for evaluating industry norms and consumer preferences.

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Most popular questions from this chapter

The sample data \(x_{1}, x_{2}, \ldots, x_{n}\) sometimes represents a time series, where \(x_{t}=\) the observed value of a response variable \(x\) at time \(t\). Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant \(\alpha\) is chosen \((0<\alpha<1)\). Then with \(\bar{x}_{2}=\) smoothed value at time \(t\), we set \(\bar{x}_{1}=x_{1}\), and for \(t=2,3, \ldots, n, \bar{x}_{t}=\alpha x_{t}+(1-\alpha) \bar{x}_{t-1}\). a. Consider the following time series in which \(x_{t}=\) temperature \(\left({ }^{\circ} \mathrm{F}\right)\) of effluent at a sewage treatment plant on day \(t: 47,54,53,50,46,46,47,50\), \(51,50,46,52,50,50\). Plot each \(x_{r}\) against \(t\) on a two-dimensional coordinate system (a time-series plot). Does there appear to be any pattern? b. Calculate the \(\bar{x}_{t} s\) using \(\alpha=.1\). Repeat using \(\alpha=.5\). c. Substitute \(\bar{x}_{r-1}=\alpha x_{r-1}+(1-\alpha) \bar{x}_{r-2}\) on the righthand side of the expression for \(\bar{x}_{l}\), then substitute \(\bar{x}_{x-2}\) in terms of \(x_{t-2}\) and \(\bar{x}_{t-3}\), and so on. On how many of the values \(x_{r} x_{t-1}, \ldots, x_{1}\) does \(\bar{x}_{t}\) depend? What happens to the coefficient on \(x_{t-k}\) as \(k\) increases? d. Refer to part (c). If \(t\) is large, how sensitive is \(\bar{x}_{1}\) to the initialization \(\bar{x}_{1}=x_{1}\) ? Explain. [Note: A relevant reference is the article "Simple Statistics for Interpreting Environmental Data," Water Pollution Control Fed. J., 1981: 167-175.]

Consider numerical observations \(x_{1}, \ldots, x_{n^{-}}\)- It is frequently of interest to know whether the \(x_{i}\) s are (at least approximately) symmetrically distributed about some value. If \(n\) is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram. However, if \(n\) is not very large, such pictures are not particularly informative. Consider the following alternative. Let \(y_{1}\) denote the smallest \(x_{p}, y_{2}\) the second smallest \(x_{i}\), and so on. Then plot the following pairs as points on a two-dimensional coordinate system: \(\left(y_{n}-\tilde{x}, \tilde{x}-y_{1}\right)\), \(\left(y_{n-1}-\tilde{x}, \tilde{x}-y_{2}\right),\left(y_{n-2}-\tilde{x}, \tilde{x}-y_{3}\right), \ldots\) There are \(n / 2\) points when \(n\) is even and \((n-1) / 2\) when \(n\) is odd. a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it (a long upper tail)? 83. Consider numerical observations \(x_{1}, \ldots, x_{n^{-}}\)- It is frequently of interest to know whether the \(x_{i}\) s are (at least approximately) symmetrically distributed about some value. If \(n\) is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram. However, if \(n\) is not very large, such pictures are not particularly informative. Consider the following alternative. Let \(y_{1}\) denote the smallest \(x_{p}, y_{2}\) the second smallest \(x_{i}\), and so on. Then plot the following pairs as points on a two-dimensional coordinate system: \(\left(y_{n}-\tilde{x}, \tilde{x}-y_{1}\right)\), \(\left(y_{n-1}-\tilde{x}, \tilde{x}-y_{2}\right),\left(y_{n-2}-\tilde{x}, \tilde{x}-y_{3}\right), \ldots\) There are \(n / 2\) points when \(n\) is even and \((n-1) / 2\) when \(n\) is odd. a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it (a long upper tail)?

The value of Young's modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations ("Strength and Modulus of a Molybdenum-Coated Ti-25Al10Nb-3U-1Mo Intermetallic," \(J\). of Materials Engr. and Performance, 1997: 46-50): \(\begin{array}{lllll}116.4 & 115.9 & 114.6 & 115.2 & 115.8\end{array}\) a. Calculate \(\bar{x}\) and the deviations from the mean. b. Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation. c. Calculate \(s^{2}\) by using the computational formula for the numerator \(S_{x x}\) d. Subtract 100 from each observation to obtain a sample of transformed values. Now calculate the sample variance of these transformed values, and compare it to \(s^{2}\) for the original data.

The following categories for type of physical activity involved when an industrial accident occurred appeared in the article "Finding Occupational Accident Patterns in the Extractive Industry Using a Systematic Data Mining Approach" (Reliability Engr. and System Safety, 2012: 108-122): A. Working with handheld tools B. Movement C. Carrying by hand D. Handling of objects E. Operating a machine F. Other Construct a frequency distribution, including relative frequencies, and histogram for the accompanying data from 100 accidents (the percentages agree with those in the cited article): \(\begin{array}{llllllllllllll}\mathrm{A} & \mathrm{B} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{F} & \mathrm{C} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{E} & \mathrm{B} & \mathrm{A} & \mathrm{C} \\ \mathrm{F} & \mathrm{D} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{A} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{E} & \mathrm{B} & \mathrm{C} & \mathrm{E} & \mathrm{A} \\ \mathrm{B} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{C} & \mathrm{D} & \mathrm{F} & \mathrm{D} & \mathrm{B} & \mathrm{B} & \mathrm{A} & \mathrm{F} \\\ \mathrm{C} & \mathrm{B} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{E} & \mathrm{B} & \mathrm{D} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{E} & \mathrm{A} & \mathrm{A} \\ \mathrm{F} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{B} & \mathrm{D} & \mathrm{C} & \mathrm{A} & \mathrm{F} & \mathrm{A} & \mathrm{A} & \mathrm{B} \\\ \mathrm{D} & \mathrm{B} & \mathrm{A} & \mathrm{E} & \mathrm{D} & \mathrm{B} & \mathrm{C} & \mathrm{A} & \mathrm{F} & \mathrm{A} & \mathrm{C} & \mathrm{D} & \mathrm{D} & \mathrm{A} \\ \mathrm{A} & \mathrm{B} & \mathrm{A} & \mathrm{F} & \mathrm{D} & \mathrm{C} & \mathrm{A} & \mathrm{C} & \mathrm{B} & \mathrm{F} & \mathrm{D} & \mathrm{A} & \mathrm{E} & \mathrm{A} \\ \mathrm{C} & \mathrm{D} & & & & & & & & & & & & \end{array}\)

For each of the following hypothetical populations, give a plausible sample of size 4 : a. All distances that might result when you throw a football b. Page lengths of books published 5 years from now c. All possible earthquake-strength measurements (Richter scale) that might be recorded in California during the next year d. All possible yields (in grams) from a certain chemical reaction carried out in a laboratory
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