Question

the top figure shows a set of linear relationships among three variables.

the bottom figure shows four path models (a through d). each variable is represented by a solid black box. an arrow connecting one box to another indicates a relationship between two variables. the sign above an arrow (+ or -) indicates whether the variables are positively or negatively related.

which path model accurately describes the relationships among the three variables?

a. model a
b. model b
c. model c
d. model d
e. i do not know the answer.

Explanation:

Analyze the linear relationships

Using the Linear Relationships knowledge point:

• The first graph shows the relationship between Variable \(x\) (horizontal axis) and Variable \(y\) (vertical axis). The line slopes downward from left to right, which indicates a negative linear relationship. Therefore, the path from \(x\) to \(y\) should have a minus sign (\(-\)).
• The second graph shows the relationship between Variable \(y\) (horizontal axis) and Variable \(z\) (vertical axis). The line slopes upward from left to right, which indicates a positive linear relationship. Therefore, the path from \(y\) to \(z\) should have a plus sign (\(+\)).

Evaluate the path models

Using the Path Models and Causal Diagrams knowledge points:

• We need a model where:
• The arrow from \(x\) to \(y\) is labeled with a minus sign (\(-\)).
• The arrow from \(y\) to \(z\) is labeled with a plus sign (\(+\)).
• Let's examine the choices shown in the bottom figure:
• Model A: Arrow \(x

ightarrow y\) has a \(+\); arrow \(y
ightarrow z\) has a \(+\).

• Model B: Arrow \(x

ightarrow y\) has a \(-\); arrow \(y
ightarrow z\) has a \(-\).

• Model C: Arrow \(x

ightarrow y\) has a \(+\); arrow \(y
ightarrow z\) has a \(-\).

• Model D: Arrow \(x

ightarrow y\) has a \(-\); arrow \(y
ightarrow z\) has a \(+\).

Select the correct model

Comparing our derived signs with the models:

• \(x

ightarrow y\) is negative (\(-\)).

• \(y

ightarrow z\) is positive (\(+\)).

• This matches Model D perfectly.

Answer:

• a. Model A
• b. Model B
• c. Model C
• d. Model D (Correct answer)
• e. I do not know the answer.