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The following frequency distribution table gives the age distribution of drivers who were at fault in auto accidents that occurred during a 1 -week period in a city. $$ \begin{array}{lr} \hline \text { Age (years) } & \boldsymbol{f} \\ \hline \text { 18 to less than } 20 & 7 \\ 20 \text { to less than } 25 & 12 \\ 25 \text { to less than } 30 & 18 \\ 30 \text { to less than } 40 & 14 \\ 40 \text { to less than } 50 & 15 \\ 50 \text { to less than } 60 & 16 \\ 60 \text { and over } & 35 \\ \hline \end{array} $$ a. Draw a relative frequency histogram for this table. b. In what way(s) is this histogram misleading? c. How can you change the frequency distribution so that the resulting histogram gives a clearer picture?

Step by step solution

01

Calculate Relative Frequency

Relative frequency is the percentage of the total count that is made up by each category in the data set. This can be calculated by dividing each original frequency \( f \) by the total sum of all frequencies in the provided data set.

02

Create Relative Frequency Table

Utilize the calculated relative frequencies to produce a new frequency table. The rows of the table will represent the age groupings and the columns will represent the relative frequencies.

03

Draw a Relative Frequency Histogram

Plot the relative frequency histogram using the relative frequency table created in Step 2. The age groups will be represented on the X-axis and the relative frequencies on the Y-axis.

04

Analyze The Histogram

Take a close look at the histogram. Note whether all age groups cover the same range of age years in the data set. If some age groups contain more years than others, this might make the histogram misleading.

05

Redraw With Equal Age Groups

In order to fix the misleading histogram from Step 4, redraw the histogram with equal age groups if they were unequal before. This gives a fairer representation of the data as each age group covers the same number of years.

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Key Concepts

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

Understanding Frequency Distribution

Frequency distribution is a method used to organize and summarize data. In a given dataset, it shows how often each different value in a set occurs. For example, the age distribution of drivers who were at fault in auto accidents can be shown in a frequency distribution table. This table categorizes data for each age group and counts how many individuals fall into each category.

• This helps researchers see patterns or trends in data at a glance.
• The frequency distribution allows us to swiftly understand the occurrence and concentration of ages in the dataset.

By putting ages into intervals (like 18-20 years, 20-25 years), the dataset becomes more manageable, allowing for easier analysis and understanding of the patterns within the data.

Calculating and Using Relative Frequency

Relative frequency gives a deeper insight than a simple frequency count by providing the proportion of the total number of items that fall within each category. To find the relative frequency, divide the frequency of each age group by the total number of entries.

• For instance, if there are 117 drivers whose ages are being analyzed, and 7 of them are aged 18-20, the relative frequency is coded as \( \frac{7}{117} \).
• This provides a fractional representation of data, making it easier to compare categories of different sizes.

Relative frequency can be expressed as a proportion, percentage, or fraction. This information is essential for drawing the relative frequency histogram, allowing for easy comparison among various groups within the dataset.

The Power of Data Visualization

Data visualization is an essential tool that transforms complex data into visual content. Creating a relative frequency histogram allows us to visually see the distribution of data across different age groups.

• A histogram displays data using bars of different heights to represent the relative frequency of each age interval.
• On the X-axis, you see different age categories, and on the Y-axis, you see the relative frequency of each group.

This visual tool is highly useful in statistics because it provides a clear and concise picture of how data is distributed over a given range. Easy interpretation is key especially in drawing quick insights from datasets.

Identifying Misleading Representations in Statistics

Statistics can sometimes be misleading if not presented accurately. In the context of a relative frequency histogram, it's crucial to ensure that each age group covers the same number of years.

• Unequal age groups can skew interpretations, as they misrepresent the data by appearing to emphasize or minimize a trend that isn’t factual.
• For example, a category that spans over 10 years will naturally have more individuals than a category spanning just 5 years, regardless of incident rates.

By adjusting the data so each age category covers the same number of years, you ensure a fair representation of the data. This helps keep conclusions drawn from the statistical analysis accurate and dependable.