Explanation
The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. The key information given is:
- When X increases by 4 units, Y increases by 1 unit
- This relationship is consistent ("whenever")
Key Concepts:
- Correlation coefficient (r) ranges from -1 to +1
- Perfect positive correlation (r = +1.00) means that for every unit change in X, there is a consistent, proportional change in Y
- The correlation coefficient is not affected by the scale of the relationship, only by its consistency
Analysis:
- The statement describes a perfect linear relationship between X and Y
- For every 4-unit increase in X, Y increases by exactly 1 unit
- This is a constant ratio: ΔY/ΔX = 1/4 = 0.25
- However, correlation measures how consistently the relationship holds, not the slope
- Since the relationship is perfectly consistent ("whenever"), the correlation is perfect
Why not 0.25?
- 0.25 would be the slope of the regression line (b = ΔY/ΔX = 1/4 = 0.25)
- But correlation (r) is different from slope
- Correlation measures the strength of the linear relationship, which is perfect here
Mathematical Relationship:
The correlation coefficient r is related to the slope b by:
r = b × (σₓ/σᵧ)
Where:
- b = slope = 0.25
- σₓ = standard deviation of X
- σᵧ = standard deviation of Y
For a perfect linear relationship, r = ±1 regardless of the slope value.
Conclusion:
The consistent, proportional relationship described indicates a perfect positive linear correlation, so r = 1.00.
