Question

1.11 Jeans A manufacturer of jeans has plants in California, Arizona, and Texas. A group of 25 pairs of jeans is randomly selected from the computerized database, and the state in which each is produced is recorded: \(\begin{array}{lllll}\text { CA } & \text { AZ } & \text { AZ } & \text { TX } & \text { CA } \\ \text { CA } & \text { CA } & \text { TX } & \text { TX } & \text { TX } \\ \text { AZ } & \text { AZ } & \text { CA } & \text { AZ } & \text { TX } \\ \text { CA } & \text { AZ } & \text { TX } & \text { TX } & \text { TX }\end{array}\) CA a. What is the experimental unit? b. What is the variable being measured? Is it qualitative or quantitative? c. Construct a pie chart to describe the data. d. Construct a bar chart to describe the data. e. What proportion of the jeans are made in Texas? f. What state produced the most jeans in the group? g. If you want to find out whether the three plants produced equal numbers of jeans, or whether one produced more jeans than the others, how can you use the charts from parts \(\mathrm{c}\) and \(\mathrm{d}\) to help you? What conclusions can you draw from these data?

Short answer

Answer: The experimental unit is the individual jeans pair selected from the three plants, and the variable being measured is the state where each jean is produced: California (CA), Arizona (AZ), or Texas (TX).

Step-by-step solution

Identifying the experimental unit

The experimental unit is the individual jeans pair selected from the three plants.

Identifying variable being measured

The variable being measured is the state where each jean is produced (CA, AZ, or TX). It is a qualitative (categorical) variable since it only describes the category but does not have a numerical value.

Constructing a pie chart

First, count the number of jeans pairs produced in each state: - CA: 8 - AZ: 7 - TX: 10 Next, calculate the percentage of jeans pairs produced in each state: - CA: (8/25) * 100 = 32% - AZ: (7/25) * 100 = 28% - TX: (10/25) * 100 = 40% Now, using these percentages, we can create a pie chart showing proportionally the number of jeans produced in each state.

Constructing a bar chart

Using the number of jeans pairs produced in each state: - CA: 8 - AZ: 7 - TX: 10 Create a bar chart with the state (CA, AZ, or TX) on the x-axis and the number of jeans pairs on the y-axis.

Proportion of jeans made in Texas

To find the proportion of jeans made in Texas, divide the number of jeans pairs produced in Texas by the total number of jeans pairs. Proportion: TX/Total = 10/25 = 0.4. So, 40% of the jeans are made in Texas.

State that produced the most jeans

Based on the counts we found: - CA: 8 - AZ: 7 - TX: 10 Texas produced the most jeans, with 10 pairs.

Using charts to draw conclusions

To find out whether the three plants produced equal numbers of jeans or whether one produced more jeans than the others, we can look at the pie chart and bar chart. Both charts show that the distribution is not equal among the three states, with Texas producing more jeans than the other two states. Based on the data given, one can conclude that the jeans production is not evenly distributed among the three plants and Texas produces more jeans than California and Arizona in the group of 25 selected jeans pairs. However, this is just a sample and may not represent the actual production ratio across the entire dataset or population.

Key concepts

Experimental Unit

In statistics, the term experimental unit refers to the smallest entity to which a treatment is applied and from which data is collected in an experiment. It's critical for accurately assessing the effects of various conditions on subjects within an experiment. In the context of the jeans manufacturing example, each individual pair of jeans represents an experimental unit. It's essential to understand this concept because the validity of an analysis can depend heavily on the definition of the experimental unit. In practical terms, when a student is analyzing data, recognizing the experimental unit helps to ensure that the correct level of analysis is performed and to avoid issues like pseudo-replication, which occurs when the independence of observations is violated.

Categorical Data Analysis

When we deal with categorical data, we are analyzing data that can be divided into groups or categories. Unlike quantitative data, categorical data is not numerical; it represents characteristics such as a product type or a demographic trait like gender or state of production as shown in the jeans example. The variable being measured in our scenario is the state (CA, AZ, or TX), which is a qualitative variable.

Categorical data analysis encompasses various statistical methods used to analyze categorical variables, including chi-square tests, logistic regression, and goodness-of-fit tests. These methods help to understand relationships between different categories, test hypotheses, and make predictions. It's important for students to grasp categorical data analysis as it is widely applicable in many fields, from marketing to medicine.

Data Visualization

The power of data visualization lies in its ability to turn raw data into a visual context, such as charts or graphs, that can be easily understood and interpreted. Visual tools like pie charts and bar charts offer clear visual comparisons between categories. For instance, the pie chart in our jeans example provides an immediate sense of proportion—how much of the whole each category represents—while the bar chart offers direct comparisons of quantities.

Good data visualization helps in uncovering patterns, revealing insights, and supporting decision-making processes. In the educational context, it's a powerful tool for learning as it aids students in grasping complex insights through visual means. Using the bar and pie charts, one can quickly determine which state has the highest and lowest production of jeans without performing complex calculations, demonstrating how visual interpretation can often be as valuable as numerical analysis in understanding data.