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) Apples: weight in grams, weight in ounces b) Apples: circumference (inches), weight (ounces) c) College freshmen: shoe size, grade point average d) Gasoline: number of miles you drove since filling up, gallons remaining in your tank

Short answer

a) Weight in grams as explanatory, expect perfect positive linear. b) Circumference as explanatory, expect positive correlation. c) Little correlation; choice irrelevant. d) Miles driven as explanatory, expect strong negative correlation.

Step-by-step solution

Understanding Variables Pair a

In the pair (a) Apples: weight in grams and weight in ounces, both variables represent the weight of apples but in different units. The explanatory variable can be one of the weight measurements, such as grams, and ounces would be the response. This is because ounces can be calculated from grams using a conversion factor.

Scatterplot Characteristics for Pair a

The scatterplot for pair (a) should show a perfect linear relationship, typically positive, since the two metrics are linearly related by a constant conversion factor. The form will be a straight line passing through the origin, and the strength will be perfect, indicating a strong association.

Understanding Variables Pair b

For pair (b), the circumference of apples could reasonably be used as an explanatory variable, while weight could be the response variable. This is because generally, as the circumference of an apple increases, its weight is expected to increase as well.

Scatterplot Characteristics for Pair b

We would expect to see a positive direction in the scatterplot for pair (b), with a form that is loosely linear or potentially curvilinear, reflecting the natural growth of apples. The strength might vary but is expected to be moderate to strong, showing noticeable correlation.

Understanding Variables Pair c

For (c), shoe size and grade point average (GPA) are largely unrelated; however, shoe size could be the explanatory variable. There is no logical cause-effect relationship, so the choice is almost arbitrary and meant to explore the lack of association.

Scatterplot Characteristics for Pair c

The scatterplot for pair (c) would likely show no strong correlation, with a form that is random or scattered with no discernible pattern. The strength of any relationship is expected to be weak to nonexistent.

Understanding Variables Pair d

In the case of (d), the number of miles driven since the last fill-up can be considered the explanatory variable, and the gallons of gasoline remaining as the response variable. This is because the miles driven affect how much gasoline is left in the tank.

Scatterplot Characteristics for Pair d

For pair (d), expect a negative direction in the scatterplot as more miles driven result in fewer gallons remaining. The form will likely be linear or near-linear with a strong negative strength indicating a clear inverse relationship.

Key concepts

Explanatory Variable

In scatterplot analysis, an explanatory variable is the one that explains or influences changes in another variable. It is essentially the independent variable, which you might expect to bring about changes in the other variable, known as the response variable.
Here's how to understand this concept better:

• In pair (a) regarding apple weights, the explanatory variable could be chosen as grams because it's the initial measurement that helps convert to ounces using a conversion factor.
• In pair (b), the circumference of an apple serves well as the explanatory variable, as changes in circumference often correlate with changes in weight.
• For pair (c), shoe size does not really affect GPA, so it's selected more to demonstrate a lack of relationship rather than as a cause.
• In pair (d), the number of miles driven since the last fill-up influences how much gasoline is left, making it a natural explanatory variable.

Response Variable

A response variable, also known as the dependent variable, reacts to or depends on changes in the explanatory variable. Its role is to help determine what effect the explanatory variable is causing. In most analyses, this connection is key to drawing conclusions.
Consider these pairs:

• For pair (a), ounces respond to changes in grams, showing how one weight measure converts to another.
• In pair (b), the weight of an apple is seen responding to changes in its circumference, as larger apples are often heavier.
• In (c), GPA doesn’t really respond to shoe size in any meaningful way, highlighting that not all explanatory-response pairs show clear causation.
• For pair (d), gallons of gasoline left in the tank respond to how many miles have been driven, illustrating consumption patterns.

Correlation Strength

Understanding correlation strength is vital when looking at scatterplots. It indicates how closely related the two variables are. It is usually quantified by the correlation coefficient, ranging from -1 to 1.
Here's a breakdown of correlation strength for each variable pair:

• For pair (a), expect a perfect correlation since grams and ounces are linearly related by a constant factor. This strong positive correlation is due to the direct conversion factor between the units.
• In pair (b), the relationship between an apple’s circumference and weight could have a moderate to strong positive correlation, depending on the natural variation in apple sizes.
• Pair (c) demonstrates a weak or negligible correlation because shoe size is unlikely to have any relationship with GPA, indicating little to no association.
• Lastly, pair (d) should show a strong negative correlation, as more miles driven result in fewer gallons of fuel, clearly depicting an inverse relationship.

Understanding these correlations helps in predicting one variable based on another and understanding their relationship.