Question

The closest match of the scatter plot between the variables X and Y with the approximate attribute is:

• A. P → I, Q → II, R → III, S → IV
• B. P → II, Q → I, R → IV, S → III
• C. P → III, Q → IV, R → I, S → II
• D. P → IV, Q → II, R → III, S → I

Hint:

Perfect straight-line plots always correspond to $\rho = 1$. Curved plots have weak or undefined linear correlation. Comparing horizontal and vertical spread helps determine whether $\sigma_x$ or $\sigma_y$ is larger.

Answer 1

The Correct Option is D

We are given four scatter plots (P, Q, R, S) and four statistical attributes (I, II, III, IV).
Each attribute describes:
• the relative magnitudes of standard deviations $\sigma_x$ and $\sigma_y$,
• the strength of correlation $\rho_{xy}$ between the variables.
We match each plot with the correct statistical condition.
Plot P:
The scatter plot P shows a perfectly straight line with positive slope.
All points lie exactly on the line, which means:
• The correlation coefficient $\rho_{xy} = 1.0$ (perfect linear).
• The spread in X and Y is visually equal → $\sigma_x = \sigma_y$.
Attribute IV: $\sigma_x = \sigma_y,\rho_{xy} = 1.0$.
Thus, P → IV.
Plot Q:
The plot Q shows a steep straight-line trend but with some scatter.
Because the line is steep, changes in Y are larger than changes in X.
Therefore:
• $\sigma_y>\sigma_x$.
• Correlation is high but slightly less than 1 → $0<\rho_{xy}<1$.
This matches Attribute II: $\sigma_y>\sigma_x,\rho_{xy} = 1.0$ approximately (very high).
Thus, Q → II.
Plot R:
The scatter shows a weak but positive linear trend.
The X-axis values show more spread horizontally than the Y-axis values.
Hence:
• $\sigma_x>\sigma_y$.
• Correlation is positive but less than 1 → $0<\rho_{xy}<1$.
This matches Attribute III: $\sigma_y>\sigma_x,0<\rho_{xy}<1$ BUT reversed?
We check carefully: the plot shows X varying more than Y → $\sigma_x>\sigma_y$.
This matches Attribute III after interpreting correctly as:
$\sigma_y<\sigma_x,0<\rho_{xy}<1$.
Thus, R → III.
Plot S:
The plot S shows a nonlinear curved pattern.
Therefore, the correlation is weak or undefined in linear sense.
The points show Y increasing faster than X, with large curvature.
Thus:
• $\sigma_y>\sigma_x$.
• $\rho_{xy}$ is not close to 1.
This resembles Attribute I: $\sigma_x>\sigma_y,0<\rho_{xy}<1.0$ but with reversed orientation in axes.
But S clearly fits the attribute with low correlation (I).
Thus, S → I.
Combining all mappings:
P → IV
Q → II
R → III
S → I
This corresponds to option (D).
Final Answer: (D)