91Ó°ÊÓ

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?

Step by step solution

01

Organize Data for Stem-and-Leaf Plot

First, sort the data in ascending order: 325, 325, 334, 339, 356, 356, 359, 359, 363, 364, 364, 366, 369, 370, 373, 373, 374, 375, 389, 392, 393, 394, 397, 402, 403, 424. Next, create the stem-and-leaf plot with stems representing the hundreds and tens digits, and leaves representing the units digit.

02

Create Stem-and-Leaf Plot

The stem-and-leaf plot is as follows: - 32 | 5 5 9 - 33 | 4 9 - 35 | 6 6 9 9 - 36 | 3 4 4 6 9 - 37 | 0 3 3 4 5 - 38 | 9 - 39 | 2 3 4 7 - 40 | 2 3 - 42 | 4 This plot suggests that the mean might be slightly larger than the median due to the skewness on the higher side.

03

Calculate Median

For the median, find the middle value in the sorted list of 26 numbers. The median is the average of the 13th and 14th values, which are 369 and 370. Thus, \(\tilde{x} = \frac{369+370}{2} = 369.5 \).

04

Calculate Mean

The mean is calculated by dividing the sum of all times by the number of data points. With a sum given as 9638:\(\bar{x} = \frac{9638}{26} \approx 370.69 \).

05

Determine Adjustments for Median (Part c)

Since the median is the average of the 13th and 14th sorted values (369 and 370), the 14th value, 370, can increase indefinitely without changing the median's position, provided it remains smaller than the 15th value (373). Similarly, the largest value, 424, can decrease down to 370 without affecting the median.

06

Convert Times to Minutes (Part d)

Convert each time from seconds to minutes by dividing by 60. Hence, the mean time in minutes is:\(\bar{x}_{minutes} = \frac{370.69}{60} \approx 6.18 \). Similarly, the median in minutes is:\(\tilde{x}_{minutes} = \frac{369.5}{60} \approx 6.16 \).

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Key Concepts

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

Stem-and-Leaf Plot

A stem-and-leaf plot is a valuable tool in descriptive statistics for visualizing the distribution and organization of data sets. It is particularly useful for small to moderate-sized data sets. In a stem-and-leaf plot, each number in the data set is split into two components: a "stem," which consists of the leading digits, and a "leaf," consisting of the final digit. For instance, with the data set given in the exercise, numbers range from 325 to 424. Here, the stems could correspond to the hundreds and tens digits, while the leaves represent the units digit. The plot then looks like this:

• 32 | 5 5 9
• 33 | 4 9
• 35 | 6 6 9 9
• 36 | 3 4 4 6 9
• 37 | 0 3 3 4 5
• 38 | 9
• 39 | 2 3 4 7
• 40 | 2 3
• 42 | 4

This plot not only displays individual data points but also suggests information about the data's distribution, such as the skewness, central tendency, and spread.

Sample Mean

The sample mean is an important measure of central tendency. It is calculated by summing all the data points in the sample and then dividing by the number of data points. The formula for calculating the sample mean \(\bar{x}\) for a data set is:\[\bar{x} = \frac{\Sigma x_{i}}{n}\]where \(\Sigma x_{i}\) is the sum of all data points and \(n\) is the number of data points. In our exercise, the sum of the data points is 9638, and there are 26 data points, so:\[\bar{x} = \frac{9638}{26} \approx 370.69\]The sample mean gives us an average value around which the data points are centered. However, it can be influenced by extreme values (outliers) as the mean considers every data point in the calculation. This can be observed in our stem-and-leaf plot, where higher values slightly skew the mean upwards.

Sample Median

The sample median is another measure of central tendency and provides a better idea of the "middle" of a data set by being less affected by outliers. To find the median in a dataset:

• Sort the data in ascending order.
• If there is an odd number of data points, the median is the middle number.
• If there is an even number of data points, the median is the average of the two middle numbers.

For our 26 data points, the median is the average of the 13th and 14th numbers in the sorted list, which are 369 and 370:\[\tilde{x} = \frac{369 + 370}{2} = 369.5\]Because the median depends only on the middle values, it is a robust measure of central tendency, particularly useful for skewed distributions.

Data Conversion

Data conversion involves changing the units of measurement of data points to other forms for analysis or comparison. Here, we convert time from seconds to minutes to understand it better in terms of real-world applications. This conversion is done by dividing each data point by 60, which transforms: \[\bar{x}_{minutes} = \frac{370.69}{60} \approx 6.18\]\[\tilde{x}_{minutes} = \frac{369.5}{60} \approx 6.16\]Performing data conversion helps in interpreting data meaningfully in contexts where a minute-based representation might be more intuitive.

Sample Data Analysis

Sample data analysis involves examining and processing data from a sample to draw conclusions or make inferences about the population from which the sample is drawn. This comprises various statistical methods, including calculating the mean and median, as well as visual representations like stem-and-leaf plots.

• First, analyze the spread and shape of data using visual tools.
• Calculate central tendency measures to summarize the data.
• Interpret these statistics to understand the dataset’s characteristics.

In our example, we noted that while both the median and mean provide insight into the central tendency of the sample, the skewness in the data affects the mean more substantially. Understanding the differences between these measures informs decisions about the sample and can highlight any significant outliers or data patterns in the sample.