91Ó°ÊÓ

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time \((\mathrm{sec})\) to complete the escape \(\left({ }^{*} \mathrm{Oxygen}\right.\) Consumption and Ventilation During Escape from an Offshore Platform," Ergonomics, 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: \(\sum x_{i}=9638 .\) ] 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 \(\bar{x}\) when the observations are reexpressed in minutes?

Step by step solution

01

Organize the Data for Stem-and-Leaf

First, let's organize the given data into a systematic form that helps display it as a stem-and-leaf plot. The data is as follows: 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. To prepare for the stem-and-leaf plot, 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.

02

Construct Stem-and-Leaf Display

Create a stem-and-leaf plot using the sorted data. The 'stem' represents the leading digits, and the 'leaf' represents the last digit. - 32 | 5, 5 - 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 display suggests that the data is slightly skewed right, indicating the mean might be larger than the median.

03

Calculate the Sample Mean

To find the sample mean, use the given hint for the sum of all observations, \(\sum x_i = 9638\). There are 26 observations.The formula for the sample mean \(\bar{x}\) is:\[\bar{x} = \frac{\sum x_i}{n}\]Where \(n\) is the number of observations.\[\bar{x} = \frac{9638}{26} = 370.69\]

04

Calculate the Sample Median

The sample median is the middle value of a sorted data set. For 26 observations, the median is the average of the 13th and 14th values.From our sorted data:- The 13th and 14th values are 369 and 370.Calculate the median:\[\text{Median} = \frac{369 + 370}{2} = 369.5\]

05

Determine the Largest Time Change for the Median

To determine by how much the largest time (424) can be changed without affecting the median 369.5, note that the median is determined by the 13th and 14th values in the sorted data. 1. **Increase**: The largest value can be increased without limit as long as the ordering up to the 13th and 14th values remains unchanged. 2. **Decrease**: Decrease 424 to any value greater than or equal to 374.5 (next highest half integer) without affecting the median.

06

Convert Times to Minutes

Convert the original data set from seconds to minutes by dividing each value by 60. Calculate the new mean and median:- For the mean in minutes:\[\bar{x} = \frac{370.69}{60} = 6.1782 \, \text{minutes}\]- For the median in minutes:\[\text{Median} = \frac{369.5}{60} = 6.1583 \, \text{minutes}\]

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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 handy way to visualize data, especially when you're dealing with a small to moderate size sample. It's similar to a histogram but maintains the original data points. The 'stem' consists of all digits except the last one, which becomes the 'leaf'.
For example, in our data set, numbers like 325 and 356 are split into stems of 32 and 35 with leaves of 5 and 6, respectively. This compact visual representation helps to quickly grasp the overall distribution of data and identify any patterns, such as clusters, gaps, and outliers.

In this exercise, the stem-and-leaf plot was constructed with the data sorted in ascending order. It was noted that the data appears slightly skewed to the right. This skewness often suggests that the mean might be larger than the median, as extreme values can pull the mean in the direction of the tail.

Sample Mean

The sample mean is a measure of central tendency, similar to the sample median, but calculated differently. It is the average of all data points in your sample. To find the mean, sum up all the values and divide by the number of observations.
Here's how it works for our data:

• Sum of all times: 9638 seconds
• Number of observations: 26
• Sample mean \(\bar{x} = \frac{9638}{26} = 370.69\)

The sample mean gives the central value of all data points combined, making it a good representation of the data set when the data is symmetrically distributed. However, it is susceptible to skewness and outliers.

Sample Median

The sample median is another measure of central tendency. Unlike the mean, it is resistant to outliers and skewness because it focuses on the middle of the dataset.
For the median in a sorted data set, find the 'middle' value. If the number of data points is odd, it is the exact middle one. If even, as in our exercise, the median is the average of the two middle numbers.
For our data, being:

• Sorted set of 26 values.
• The 13th and 14th values are 369 and 370.
• Median = \(\frac{369 + 370}{2} = 369.5\)

This shows the robustness of the median against any extreme values, making it a reliable measure when dealing with skewed distributions.

Data Re-expression

Data re-expression involves converting data into a different unit or form, which can help to simplify or clarify analysis. Here, the data on escape times is originally in seconds, and re-expressing them in minutes can make the interpretation easier, especially when considering human activities where minutes are more intuitive.
To re-express the data in minutes, divide each value by 60. This transformation affects the mean and median:

• Mean in minutes: \( \frac{370.69}{60} = 6.1782 \) minutes
• Median in minutes: \( \frac{369.5}{60} = 6.1583 \) minutes

Data re-expression can help in standardizing the values for comparison with other datasets or adapting to practical measurement units. It's useful in making data more interpretable or in adjusting skewness through methods like logarithmic transformations.