91Ó°ÊÓ

Consider the following data on type of health complaint \((\mathrm{J}=\) joint swelling, \(\mathrm{F}=\) fatigue, \(\mathrm{B}=\) back pain, \(\mathrm{M}=\) muscle weakness, \(\mathrm{C}=\) coughing, \(\mathrm{N}=\) nose running/irritation, \(\mathrm{O}=\) other) made by tree planters. Obtain frequencies and relative frequencies for the various categories, and draw a histogram. (The data is consistent with percentages given in the article "Physiological Effects of Work Stress and Pesticide Exposure in Tree Planting by British Columbia Silviculture Workers"" Ergonomics, 1993: 951–961.) $$ \begin{array}{llllllllllllll} \mathrm{O} & \mathrm{O} & \mathrm{N} & \mathrm{J} & \mathrm{C} & \mathrm{F} & \mathrm{B} & \mathrm{B} & \mathrm{F} & \mathrm{O} & \mathrm{J} & \mathrm{O} & \mathrm{O} & \mathrm{M} \\ \mathrm{O} & \mathrm{F} & \mathrm{F} & \mathrm{O} & \mathrm{O} & \mathrm{N} & \mathrm{O} & \mathrm{N} & \mathrm{J} & \mathrm{F} & \mathrm{J} & \mathrm{B} & \mathrm{O} & \mathrm{C} \\ \mathrm{J} & \mathrm{O} & \mathrm{J} & \mathrm{J} & \mathrm{F} & \mathrm{N} & \mathrm{O} & \mathrm{B} & \mathrm{M} & \mathrm{O} & \mathrm{J} & \mathrm{M} & \mathrm{O} & \mathrm{B} \\ \mathrm{O} & \mathrm{F} & \mathrm{J} & \mathrm{O} & \mathrm{O} & \mathrm{B} & \mathrm{N} & \mathrm{C} & \mathrm{O} & \mathrm{O} & \mathrm{O} & \mathrm{M} & \mathrm{B} & \mathrm{F} \\ \mathrm{J} & \mathrm{O} & \mathrm{F} & \mathrm{N} & & & & & & & & & & \end{array} $$

Step by step solution

01

Count Frequencies

First, identify each unique category in the data: Joint swelling (J), Fatigue (F), Back pain (B), Muscle weakness (M), Coughing (C), Nose running/irritation (N), and Other (O). Now, count how many times each category appears in the dataset. Start with category J and proceed through to category O. For example, count all the J's in the dataset, then the F's, and so on. After counting, you will have: - J appears 9 times. - F appears 8 times. - B appears 7 times. - M appears 4 times. - C appears 3 times. - N appears 5 times. - O appears 24 times.

02

Calculate Relative Frequencies

Relative frequencies are calculated by dividing the frequency of each category by the total number of data points. First, count the total number of complaints, which is 60. Then calculate the relative frequency for each category by using the formula:\[ \text{Relative Frequency} = \frac{\text{Frequency of category}}{\text{Total number of data points}} \]For example, for Joint swelling (J):\[ \text{Relative Frequency of J} = \frac{9}{60} = 0.15 \]Calculate similarly for the remaining categories:- F: \( \frac{8}{60} = 0.133 \)- B: \( \frac{7}{60} = 0.117 \)- M: \( \frac{4}{60} = 0.067 \)- C: \( \frac{3}{60} = 0.05 \)- N: \( \frac{5}{60} = 0.083 \)- O: \( \frac{24}{60} = 0.4 \)

03

Draw the Histogram

To create a histogram, first place the categories on the x-axis and frequency values on the y-axis. Draw a bar for each category reaching up to its calculated frequency. For example: - The bar for 'O' will have a height of 24. - The bar for 'J' will have a height of 9. - The bar for 'F' will have a height of 8. - Continue similarly for the other categories (B, M, C, and N). Make sure all bars are equally spaced and labeled for clarity. This histogram will visually represent the frequency of each type of health complaint among tree planters.

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Key Concepts

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

Frequency Distribution

Frequency distribution is the process of organizing raw data into categories and then counting the number of occurrences in each category. This method helps to summarize large data sets and find out how often each category appears. In the given problem, the categories are different types of health complaints experienced by tree planters, such as joint swelling, fatigue, back pain, muscle weakness, coughing, nose running/irritation, and other complaints.
To create a frequency distribution, simply tally the number of times each type of complaint appears in the dataset.

• Joint Swelling (J) - 9 times
• Fatigue (F) - 8 times
• Back Pain (B) - 7 times
• Muscle Weakness (M) - 4 times
• Coughing (C) - 3 times
• Nose Running/Irritation (N) - 5 times
• Other (O) - 24 times

Counting these frequencies gives us a practical view of the dataset, showing which health complaints are most and least common among the tree planters.

Relative Frequency

Relative frequency is a helpful concept that tells us the proportion of each category relative to the whole data set. It's calculated by dividing the frequency of each category by the total number of data points.
For example, if you have a total of 60 complaints logged and 9 of them are joint swelling, the relative frequency for joint swelling is calculated as:
\[ \frac{9}{60} = 0.15 \\] This means that 15% of the complaints were about joint swelling. Let's compute relative frequencies for all categories:

• Joint Swelling (J): 0.15
• Fatigue (F): 0.133
• Back Pain (B): 0.117
• Muscle Weakness (M): 0.067
• Coughing (C): 0.05
• Nose Running/Irritation (N): 0.083
• Other (O): 0.4

Having the relative frequencies helps in understanding the data proportionally, making it easier to compare the prevalence of one complaint to another. It's crucial for converting counts into more interpretable data that represent parts of the whole.

Histogram

A histogram is a type of bar chart that represents the distribution of a dataset. It gives a visual impression of how the data is spread across different categories. In our example, the categories are different health complaints.
To draw a histogram, place the complaints on the x-axis. Also, add the frequencies on the y-axis. Each bar will represent a complaint, and its height will correspond to its frequency.

• Joint Swelling (J) - Height of 9
• Fatigue (F) - Height of 8
• Back Pain (B) - Height of 7
• Muscle Weakness (M) - Height of 4
• Coughing (C) - Height of 3
• Nose Running/Irritation (N) - Height of 5
• Other (O) - Height of 24

This graphical representation helps quickly identify trends or patterns in the data, making it easy to see which complaints are most common at a glance. Ensure the bars are drawn with equal width and that there are spaces between each to distinguish different categories.

Categorical Data Analysis

Categorical data analysis involves examining and interpreting data where each entry belongs to a specific category and is qualitative. Categories in a dataset could be any non-numeric data, like set labels or groups. For this exercise, categorical data analysis helped us investigate health complaints among tree planters.
The first step in analyzing this type of data is categorizing and counting each entry, which allows for constructing a frequency distribution.
Following this, calculating relative frequencies further analyzes the proportion of each category within the whole dataset. This helps in making insightful conclusions on the data.
Finally, constructing a histogram provides a visual summary that enhances our ability to communicate findings effectively. Through the comprehensive steps of tallying, computing, and graphing, categorical data analysis offers a robust toolkit for interpreting qualitative datasets, such as health complaints, in an accessible and meaningful way.