The variables response time to a drug and drug dosage are linearly associated, withr = −0.9. From this information, we can conclude that:

Question

BlackTom AI · Accepted Answer

A Pearson correlation r = −0.9 indicates a strong negative linear relationship: as one variable increases, the other tends to decrease.

1. Relevant knowledge points and concepts
- The correlation coefficient r measures direction and strength of a linear relationship between two variables. 
- Sign of r: positive = both variables move in same direction; negative = they move in opposite directions. 
- Magnitude of r (0 to 1) indicates strength; |r| = 0.9 is very strong. 
- r is a standardized measure (no units) and is not the slope of a regression line; correlation ≠ causation.

2. Step-by-step reasoning
- Given r = −0.9 for response time and drug dosage, the negative sign tells us that higher dosage is associated with lower response time. 
- The magnitude 0.9 (close to 1) tells us this inverse relationship is strong and consistent across observations.

3. Calculations / intermediate steps
- Compute r^2 to quantify explained variance: r^2 = (−0.9)^2 = 0.81. 
- Interpretation: about 81% of the variability in response time (in a linear sense) is associated with variability in dosage. This confirms a strong linear association but does not change the direction indicated by the negative sign.

4. Why the correct answer is correct
- Option c states “response times decrease with increased drug dosage.” That matches the negative sign (inverse relationship) and the strong magnitude (near-perfect negative linear association). Thus it correctly describes the direction (decrease) when dosage increases.

5. Why the other options are incorrect
- a: “response times increase with increased drug dosage” is the opposite of what a negative r indicates. 
- b: “response times decrease with decreased drug dosage” implies both variables move together (decrease-decrease), which is a positive association; this contradicts r < 0. 
- d: “response times are −0.9 times the drug dosage” misinterprets r as a regression slope; r is unitless and not equal to the change in response time per unit dosage.

6. Conclusion
Because r = −0.9 denotes a very strong negative linear association (higher dosage → lower response time), the correct choice is c. This matches the provided answer.

The variables response time to a drug and drug dosage are linearly associated, withr = −0.9. From this information, we can conclude that:Single choice

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