91Ó°ÊÓ

College students who were drivers were asked if they had ever driven a car 100 mph or more (yes or no). The results are shown in the table, along with gender. a. There are two variables in the table, state what they are and whether each is categorical or numerical. b. Make a two-way table of the results with Male and Female across the top and Yes and No at the left edge. c. Compare the percentages of men and women who have driven \(100 \mathrm{mph}\) or more. $$ \begin{array}{|c|c|c|c|} \hline \text { Gender } & \mathbf{1 0 0}+\text { mph } & \text { Gender } & \mathbf{1 0 0}+\mathbf{~ m p h} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{y} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{y} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{n} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{n} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{m} & \mathrm{n} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{f} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{f} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \mathrm{f} & \mathrm{y} & \mathrm{f} & \mathrm{n} \\ \hline \end{array} $$

Step by step solution

01

Identify the Variables

From the table, the two variables can be identified as 'Gender' and '100+ mph'. The 'Gender' variable is categorical since it represents a category (Male or Female), while '100+ mph' is also categorical, representing whether the individuals did (Yes) or did not (No) drive at a speed equal to or above 100 mph.

02

Count and Prepare Data for the Two-way Table

Count the number of instances in each category. For males, there are 9 individuals in total, out of which 6 drove at 100+ mph (Yes) and 3 did not (No). For females, there are 10 individuals in total, out of which 4 drove at 100+ mph, while 6 did not.

03

Create the Two-way Table

Using the counted data, the two-way table would be as follows: \[ \begin{array}{|c|c|c|} \hline & \textbf{Male} & \textbf{Female} \ \hline \textbf{Yes} & 6 & 4 \ \hline \textbf{No} & 3 & 6 \ \hline \end{array} \]

04

Calculate the Percentages

To compare the percentages of men and women who have driven 100 mph or more, calculate percentage for each category. For males: \( (6/9)*100= 66.67% \) and for females: \( (4/10)*100= 40% \)

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Key Concepts

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

Categorical Data

When dealing with statistics, it's essential to recognize the type of data you're analyzing. Categorical data is one such type that includes values which represent categories or groups. Think of categorical data as a way to label a characteristic. For example, 'Gender' with categories such as 'Male' or 'Female', and '100+ mph' with 'Yes' or 'No' responses, as shown in our textbook problem, are both instances of categorical data. Unlike numerical data, which can be quantified and subjected to mathematical operations, categorical data is qualitative.

Understanding the nature of categorical data is crucial because it determines the statistical methods you will use for analysis. In our exercise example, identifying that both 'Gender' and '100+ mph' are categorical variables shapes the way we organize and interpret the data – leading us to use a two-way table.

Data Analysis in Statistics

In statistics, data analysis is a broad term for the various ways data can be inspected, cleaned, transformed, and modeled with the aim of discovering useful information. It's a step-by-step process that often starts with data collection and ends with drawing conclusions or making decisions based on the findings. A crucial aspect of data analysis is using the appropriate method for the type of data you have.

For instance, a two-way table is a great tool for organizing and displaying categorical data, particularly when you’re interested in the relationship between two different categories. In our exercise, the two-way table facilitates a clearer view of the count of males and females who have driven 100 mph or more. This visual representation simplifies the complex information into an easily digestible format, which is then used to perform further analysis, such as comparing percentages.

Comparing Percentages

The comparison of percentages is a common task in statistics, used to understand the relative differences between groups. In the context of our textbook problem, we compare the percentage of males to females who have driven a car at 100 mph or more. This is done to identify potential differences in behavior between genders. To obtain the percentages, we take the count of each gender who have driven at that speed, and divide it by the total number of individuals in that gender category.

The calculation for males results in approximately 66.67% and for females it's 40%. The comparison clearly shows that a higher percentage of male college students have driven at 100 mph when compared to female college students. This step is key for interpreting the two-way table and can provide insights into the behavioral patterns that may exist between different categorical groups. It is essential to provide clear and concise explanations when comparing such percentages so that students can draw meaningful conclusions from the data analysis.