The association between two variables y and x, asshown in the scatterplot above, is non-linear. In an attempt totransform the relationship to linearity, a student wouldbe advised to:

Question

BlackTom AI · Accepted Answer

The scatter shows y increasing with x but with a decreasing slope (concave down): differences in y are larger at small x and get smaller as x grows. Compressing larger y-values will tend to straighten that pattern, so a log transformation of y is appropriate.

Relevant concepts:
- A transformation can make a nonlinear association approximately linear by stretching/compressing parts of the scale.
- log(y) compresses large y values more than small ones (useful when growth slows or when multiplicative/exponential effects are present).
- y^2 magnifies large y values; 1/y inverts the relationship; omitting points is not a valid transformation.

Step-by-step reasoning with numerical illustration:
1. Read the pattern: y rises rapidly at first and then levels off (concave down).
2. Consider the effect of candidate transforms on a representative sequence of y values, e.g. y = [1, 2, 3, 4, 4.5].
   - log(y) (natural logs): [0.00, 0.693, 1.099, 1.386, 1.504] → these transformed increments are more uniform, reducing curvature.
   - y^2: [1, 4, 9, 16, 20.25] → differences become much larger, increasing curvature.
   - 1/y: [1.00, 0.500, 0.333, 0.250, 0.222] → the relationship is now decreasing with x and compressed near zero, which does not straighten the original increasing pattern.
3. Because log(y) makes the spacing between larger y-values smaller, a concave-down increasing pattern becomes closer to a straight line after the transform.

Why the correct answer is correct:
- The log transform on y reduces the impact of large y-values and tends to linearize relationships that are multiplicative or show diminishing marginal increases. The numerical example demonstrates that log(y) yields more uniform increments consistent with linearity.

Why the other options are incorrect:
- a) y^2 exaggerates large values and would worsen the curvature.
- c) Omitting the first four points is data deletion, not a legitimate transformation, and would discard information.
- d) 1/y inverts and compresses values near zero, producing a decreasing pattern and not straightening the observed increase.

Conclusion: Applying a log transformation to y is the appropriate choice; this matches the provided correct answer (b. use a log y transformation).

The association between two variables y and x, asshown in the scatterplot above, is non-linear. In an attempt totransform the relationship to linearity, a student wouldbe advised to:Single choice

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