Question

Association. Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) Long-distance calls: time (minutes), cost b) Lightning strikes: distance from lightning, time delay of the thunder c) A streetlight: its apparent brightness, your distance from it d) Cars: weight of car, age of owner

Short answer

Explanatory variables: a) time, b) distance, c) distance, d) age of owner. Likely: a) positive linear, b) positive linear, c) negative non-linear, d) no clear pattern.

Step-by-step solution

Understand the Problem

We are tasked with identifying explanatory and response variables for pairs of data, predicting scatterplot characteristics like direction, form, and strength.

Determine Variables for Pair a

For long-distance calls, time (minutes) and cost are given. The explanatory variable is usually the one that suggests cause or helps predict. Therefore, time (minutes) is the explanatory variable, and cost is the response variable.

Predict Scatterplot for Pair a

As time increases, the cost would likely increase, suggesting a positive direction (direct correlation). The form may be linear if the cost increases consistently with time. Strength would depend on factors affecting call rates.

Determine Variables for Pair b

For lightning strikes, the distance from lightning affects the time delay of the thunder. Thus, the distance is the explanatory variable and the time delay is the response variable.

Predict Scatterplot for Pair b

As the distance from a lightning strike increases, the time delay of the thunder increases, suggesting a positive direction. The form is likely linear. The strength should be strong, as the speed of sound is fairly constant.

Determine Variables for Pair c

For a streetlight, apparent brightness changes with your distance from the light. Distance from the streetlight is the explanatory variable, and apparent brightness is the response variable.

Predict Scatterplot for Pair c

As distance increases, apparent brightness decreases, suggesting a negative direction (inverse correlation). It might form a non-linear (possibly exponential decay) pattern. The strength is potentially strong since brightness diminishes rapidly with distance due to the inverse square law.

Determine Variables for Pair d

For cars, there isn't a clear causative relationship, but if choosing, the age of the owner might generically seem to influence the weight of car preference. Owner age could be the explanatory variable, while car weight is the response variable.

Predict Scatterplot for Pair d

There's no inherent correlation expected in strength or form. We might not predict any specific direction, form, or strength due to the lack of causal or predictable relationship.

Key concepts

Explanatory Variable

An explanatory variable, also known as an independent variable, is the one that is used to explain or predict another variable, known as the response variable. In the context of a scatterplot, the explanatory variable is typically plotted along the x-axis. It represents the variable that you suspect might be influencing or causing changes in another variable. For example, in long-distance phone calls, the time spent on the call (in minutes) is the explanatory variable. We hypothesize that this variable will affect the cost of the call, which would be the response variable. Choosing the right explanatory variable helps us better understand the relationship between variables and predict outcomes more accurately.

Response Variable

The response variable, or the dependent variable, is the outcome you are trying to predict or explain. It is usually plotted on the y-axis in a scatterplot. This variable changes in response to variations in the explanatory variable. Consider the example of a streetlight: as your distance from the light changes, the apparent brightness alters. Here, the apparent brightness is the response variable as it is dependent on the distance from the light. By examining how the response variable changes with different values of the explanatory variable, we can gain insights into the nature and strength of the relationship between the two variables.

Correlation Direction

Correlation direction refers to the general trend seen in a scatterplot, indicating how two variables are related. The direction can be:

• Positive (Direct Correlation): As the explanatory variable increases, the response variable also increases. An example of this is the relationship between the time spent on a phone call and its cost; more time generally means higher cost.
• Negative (Inverse Correlation): As the explanatory variable increases, the response variable decreases. For instance, as the distance from a streetlight increases, the apparent brightness decreases.
• No Correlation: There's no discernible pattern or consistent trend in how the variables are related, such as in the case of car weight versus the age of the owner. Here, no specific correlation or trend might appear because there is no inherent causative relationship.

Understanding these directions helps in interpreting the scatterplot data and predicting the relationships in other similar datasets.

Data Visualization

Data visualization involves representing data in a visual context, like a scatterplot, to make it easier to see patterns, trends, and outliers. Scatterplots are particularly useful as they display the relationship between two quantitative variables. By plotting each pair of corresponding data points from two variables, we can visually analyze how one affects the other. When visualizing data, it is important to:

• Choose appropriate scales for the axes to accurately reflect the data range.
• Label each axis with the correct variables to avoid confusion.
• Look for overall patterns, clusters, or outliers that can inform further analysis or raise questions about underlying causes.

Effective data visualization not only aids in data analysis by providing clear insights into variable relationships but also helps communicate findings in a more intuitive and impactful way.