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) When climbing mountains: altitude, temperature b) For each week: ice cream cone sales, air-conditioner sales c) People: age, grip strength d) Drivers: blood alcohol level, reaction time

Short answer

a) Altitude as explanatory, temperature as response; negative correlation. b) Could go either way, but traditionally, ice cream as explanatory; positive trend. c) Age as explanatory, grip strength as response; negative trend. d) Blood alcohol as explanatory, reaction time as response; positive correlation.

Step-by-step solution

Understand Variables

Begin by identifying the two variables in each pair. The explanatory variable is the one that explains or predicts changes in the response variable. The response variable is the one affected by changes in the explanatory variable.

Pair Analysis a

For pair (a), 'altitude' and 'temperature', consider which variable affects the other. Typically, temperature decreases with increasing altitude. Thus, altitude is the explanatory variable, while temperature is the response variable.

Pair Analysis b

For pair (b), 'ice cream cone sales' and 'air-conditioner sales', consider what drives demand. Generally, warmer weather (not explicitly stated but implied) would increase both sales. Here, either could be considered as explanatory under specific contexts, but practically, these would both be response variables to weather conditions. For simplicity, choose ice cream cone sales as explanatory for easy analysis, predicting air-conditioner sales as response.

Pair Analysis c

For pair (c), 'age' and 'grip strength', contemplate the likely influence. As people age, their grip strength tends to change, often decreasing after a certain point. Thus, age is the explanatory variable, and grip strength is the response variable.

Pair Analysis d

For pair (d), 'blood alcohol level' and 'reaction time', blood alcohol level directly impacts reaction time, making it the explanatory variable, while reaction time is the response variable.

Discussing Scatterplots

For each scenario, considering the explanatory and response variables, expect certain patterns in the scatterplot: - (a) likely shows a negative correlation (inversely related), often linear, with moderate to strong strength. - (b) might show a positive trend if temperature were included, likely non-linear, with moderate strength. - (c) likely exhibits a negative, potentially nonlinear relationship, with varying strength as age increases. - (d) shows a positive correlation, likely linear, with strong strength as higher alcohol levels tend to slow reaction time.

Key concepts

Explanatory Variable

An explanatory variable provides context or insight into the changes of another variable. It acts like a cause in a cause-and-effect relationship.
In our exercise examples, the explanatory variables were chosen based on their effects on response variables.

• For altitude and temperature, altitude is explanatory as temperature usually drops with higher altitudes.
• In the relationship between age and grip strength, age acts as the explanatory variable since grip strength often changes with age.
• Blood alcohol level is explanatory regarding reaction time, as higher levels directly affect the ability to react quickly.

Understanding which variable serves as the explanatory one can significantly aid in predicting outcomes, thereby simplifying data analysis.

Response Variable

A response variable is generally what you measure in an experiment or study and is influenced by the explanatory variable.
It can be thought of as the effect in a cause-and-effect relationship.

• In mountain climbing, temperature becomes the response variable since it's affected by the altitude.
• Grip strength is the response variable that's potentially reduced as people age.
• For driving under the influence, reaction time is the response variable because it changes as blood alcohol levels rise.

By identifying the response variable, one can better capture the impact of varying conditions, making it essential for effective data analysis and interpretation.

Correlation

Correlation describes the relationship between two variables on a scatterplot, explaining whether and how strongly they are connected.
In our exercise, different pairings show distinct types of correlation:

• Altitude and temperature display a negative correlation, meaning as one increases, the other decreases.
• Ice cream sales to air-conditioner sales might show a positive correlation if considering external factors like temperature.
• The bond between age and grip strength likely reveals a negative correlation, with grip strength reducing over time.
• Blood alcohol level and reaction time show a positive correlation, as alcohol impairs the ability to react quickly.

Understanding correlation helps predict trends and lays groundwork for creating insightful predictions.

Data Analysis

Data analysis entails examining raw data to uncover patterns or draw conclusions. In scatterplots, this involves understanding the relationship between the explanatory and response variables.
For each set of variables discussed in the exercise:

• Altitude and temperature analysis may show a strong inverse linear pattern.
• Sales data for ice cream and air-conditioners could depict a nonlinear, moderate trend when temperature is considered.
• Age versus grip strength analysis might indicate a fluctuating downward pattern, not strictly linear.
• The blood alcohol and reaction time study likely reveals a strong positive linear relationship.

By carefully analyzing these scatterplots, one can identify trends, predict outcomes, and make data-driven decisions, fueling effective research and study outcomes.