Question

A relationship between two variables is described. In each case, we can think of one variable as helping to explain the other. Identify the explanatory variable and the response variable. Year and the world record time in a marathon.

Short answer

The explanatory variable is 'Year' and the response variable is 'World record time in a marathon'.

Step-by-step solution

Identifying the Explanatory Variable

The explanatory variable is the variable which causes change in the other variable. Here, the 'year' is causing changes in the 'world record time in a marathon' as with each passing year, the world record time in marathon can improve or degrade.

Identifying the Response Variable

The response variable is the variable that is affected by the changes in the explanatory variable. Here, the 'world record time in a marathon' is being affected by the 'year'. As years pass, training methods, equipment, and understanding of human endurance improve, potentially leading to faster marathon times. Therefore, 'world record time in a marathon' is the response variable.

Key concepts

Statistical Relationships

When analyzing data, we often seek to understand the relationship between two different variables. This understanding allows us to explore how changes in one aspect of a situation might influence another. For example, in the exercise provided, the variables of 'year' and 'world record time in a marathon' are examined to establish a relationship.

In statistical terms, we call this relationship a correlation when changes in one variable are related to changes in another. This correlation can be positive, with both variables moving in the same direction, or negative, with the variables moving in opposite directions. By using scatter plots, line graphs, or statistical measures like Pearson’s correlation coefficient, we can visually and numerically describe these relationships.

Importance of Patterns

Patterns in data help us predict future trends. If there's a strong negative correlation between the 'year' and the 'world record time in a marathon,' one might predict that as years go by, record times will decrease, indicating improved performance. However, correlation does not establish causation - it simply flags patterns for further investigation.

Variable Causation

A crucial part of understanding statistical relationships is distinguishing between correlation and causation. The concept of causation indicates that one event is the result of the occurrence of the other event; there is a directional relationship.

In the textbook exercise, the 'year' is seen as the explanatory variable that potentially influences changes in the 'world record time in a marathon,' the response variable. It is hypothesized that as years pass, advancements may contribute to faster marathon times, implying a form of causation.

Establishing Causation

In order to establish causation, researchers often have to perform controlled experiments or longitudinal studies. These kinds of detailed analyses are required to rule out other variables that may affect the response variable. Hence, even if yearly improvements suggest a causal link to marathon times, it would require more rigorous studies to affirm that 'year' indeed causes changes in 'world record time in a marathon.'

Data Analysis

Data analysis encompasses a range of techniques and procedures that allow us to summarize, interpret, and draw conclusions from data. It involves cleaning, inspecting, transforming, and modeling data to discover useful information that can support decision-making.

The process of data analysis would, for instance, include identifying the explanatory and response variables as the first step towards understanding how they interact with each other. When dealing with the marathon record times and years in our example, we can use statistical software or graphing tools to observe trends and variations over time.

Contextual Understanding

It is also essential to consider the context behind the numbers. Factors such as improved training techniques, better nutrition, and technological advancements in gear could all interplay with 'year' to impact 'world record time in a marathon.' Detailed data analysis helps to isolate the variables of interest while accounting for or acknowledging the underlying factors that may confound the results.