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Chapter 1: Problem 38

A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-brake movement time, in ms) during real driving conditions at \(60 \mathrm{~km} /\) hr gave the following summary information on the distribution of times ("A Field Study on Braking Responses During Driving," Ergonomics, 1995: 1903鈥�1910): mean \(=535\) median \(=500 \quad\) mode \(=500\) \(\mathrm{sd}=96 \quad\) minimum \(=220 \quad\) maximum \(=925\) 5 th percentile \(=400 \quad 10\) th percentile \(=430\) 90 th percentile \(=640 \quad 95\) th percentile \(=720\) What can you conclude about the shape of a histogram of this data? Explain your reasoning.

Short Answer

Expert verified
The histogram is likely right-skewed.

Step by step solution

01

Compare Mean, Median, and Mode

To determine the skewness of a distribution, compare the mean, median, and mode. In this data: \( \text{mean} = 535 \), \( \text{median} = 500 \), and \( \text{mode} = 500 \). When the mean is greater than the median and mode, it often indicates a right-skewed (positively skewed) distribution.
02

Analyze Percentiles

Percentiles provide additional insights into skewness. Here, the 90th percentile is 640 and the 10th percentile is 430. The distance from the median (500) to the 90th percentile (640) is greater than to the 10th percentile (430), supporting the indication of a right skew.
03

Consider the Range and Outliers

The range, given by \( \text{maximum} = 925 \) and \( \text{minimum} = 220 \), shows the data spans 705 ms. The skewness might also be due to potential outliers on the higher end, as the maximum value is significantly higher than the mean.
04

Conclusion on Histogram Shape

The analysis suggests that the distribution is right-skewed since the mean is greater than the median and mode, the 90th percentile is farther from the median compared to the 10th percentile, and there might be outliers on the higher end of the data.

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Key Concepts

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

Understanding Skewness in Data Distributions
Skewness refers to the asymmetry in a statistical distribution. In simple terms, it tells us whether the bulk of data values lies to the right or left of the mean. A perfectly symmetrical distribution, like the normal distribution, has a skewness of zero, meaning the data is balanced on both sides. However, real-world data often show some degree of skewness.
If a distribution is right-skewed (positively skewed), the tail on the right side is longer or fatter than the left side. This means that the mean is usually greater than the median, pulling the average upward due to higher values or outliers on the right. Conversely, in a left-skewed (negatively skewed) distribution, the mean is less than the median as the tail is longer on the left.
In our exercise, the mean braking time is 535 ms, while the median and mode are both 500 ms. Because the mean is greater than both the median and mode, it indicates a right-skewed distribution. This suggests that some drivers took significantly longer to brake, pulling the mean to the right.
Percentiles and What They Reveal
Percentiles are crucial as they divide the data into 100 equal parts, helping us understand the spread and skewness of a data set. They are particularly useful in identifying the position of an observation within a distribution. When examining the braking times, the 10th percentile is 430 ms and the 90th percentile is 640 ms.
A right-skewed distribution is often distinguished by a greater gap between the upper percentiles and the median than that between the lower percentiles and the median. This is exactly what we observe since the 90th percentile is further from the median (140 ms) compared to the 10th percentile (70 ms). This asymmetric gap further supports the indication of right skewness in the braking time data. Understanding percentiles allows us to assess where the bulk of the data lies and how spread out they are around the central value.
Using Histogram Analysis to Visualize Distributions
Histograms provide a visual representation of the frequency distribution of a data set. They help in understanding the shape of the data and identifying patterns such as skewness, modality (number of peaks), and potential outliers.
In the braking time data, a histogram would likely show a peak around the median and mode of 500 ms, with a tail extending to the right, reflecting the right-skewed distribution. The range, which is the difference between the maximum (925 ms) and minimum (220 ms), suggests a wide spread of data with potential outliers on the higher end. These outliers are often responsible for the skewness in the distribution. By examining a histogram, one can visually infer the degree of skewness, the presence of outliers, and how the data clusters around central measures like the mean and median.
In summary, histogram analysis is a vital tool in statistical data analysis, allowing students to see the distribution patterns and connect them with numerical measures such as mean, median, mode, percentiles, and range.

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Most popular questions from this chapter

Every corporation has a governing board of directors. The number of individuals on a board varies from one corporation to another. One of the authors of the article "'Does Optimal Corporate Board Size Exist? An Empirical Analysis" (J. of Applied Finance, 2010: 57-69) provided the accompanying data on the number of directors on each board in a random sample of 204 corporations. \(\begin{array}{lrrrrrr}\text { No. directors: } & 4 & 5 & 6 & 7 & 8 & 9 \\\ \text { Frequency: } & 3 & 12 & 13 & 25 & 24 & 42 \\ \text { No. directors: } & 10 & 11 & 12 & 13 & 14 & 15 \\ \text { Frequency: } & 23 & 19 & 16 & 11 & 5 & 4\end{array}\) \(\begin{array}{lrrrrr}\text { No. directors: } & 16 & 17 & 21 & 24 & 32 \\ \text { Frequency: } & 1 & 3 & 1 & 1 & 1\end{array}\) a. Construct a histogram of the data based on relative frequencies and comment on any interesting features. b. Construct a frequency distribution in which the last row includes all boards with at least 18 directors. If this distribution had appeared in the cited article, would you be able to draw a histogram? Explain. c. What proportion of these corporations have at most 10 directors? d. What proportion of these corporations have more than 15 directors?

A mutual fund is a professionally managed investment scheme that pools money from many investors and invests in a variety of securities. Growth funds focus primarily on increasing the value of investments, whereas blended funds seek a balance between current income and growth. Here is data on the expense ratio (expenses as a \(\%\) of assets, from www morningstar.com) for samples of 20 large-cap balanced funds and 20 large-cap growth funds ("largecap" refers to the sizes of companies in which the funds invest; the population sizes are 825 and 762 , respectively): \(\begin{array}{llllll}\text { BI } & 1.03 & 1.23 & 1.10 & 1.64 & 1.30 \\ & 1.27 & 1.25 & 0.78 & 1.05 & 0.64 \\ & 0.94 & 2.86 & 1.05 & 0.75 & 0.09 \\ & 0.79 & 1.61 & 1.26 & 0.93 & 0.84 \\ \text { Gr } & 0.52 & 1.06 & 1.26 & 2.17 & 1.55 \\ & 0.99 & 1.10 & 1.07 & 1.81 & 2.05 \\ & 0.91 & 0.79 & 1.39 & 0.62 & 1.52 \\ & 1.02 & 1.10 & 1.78 & 1.01 & 1.15\end{array}\) a. Calculate and compare the values of \(\bar{x}, \tilde{x}\), and \(s\) for the two types of funds. b. Construct a comparative boxplot for the two types of funds, and comment on interesting features.

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape ("Oxygen Consumption and Ventilation During Escape from an Offshore Platform," Ergonomies, 1997: 281-292): \(\begin{array}{lllllllll}389 & 356 & 359 & 363 & 375 & 424 & 325 & 394 & 402 \\\ 373 & 373 & 370 & 364 & 366 & 364 & 325 & 339 & 393 \\ 392 & 369 & 374 & 359 & 356 & 403 & 334 & 397 & \end{array}\) a. Construct a stem-and-leaf display of the data. How does it suggest that the sample mean and median will compare? b. Calculate the values of the sample mean and median. [Hint: \(\left.\Sigma x_{i}=9638 .\right]\) c. By how much could the largest time, currently 424 , be increased without affecting the value of the sample median? By how much could this value be decreased without affecting the value of the sample median? d. What are the values of \(\bar{x}\) and \(\tilde{x}\) when the observations are reexpressed in minutes?

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