Sensitivity (Meta) Math Example 4

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Example 4

medium
A model predicts exam score S=10log⁡10(h)S = 10\log_{10}(h) where hh is hours of study. If hh changes from 10 to 11 hours, compute the change in SS and discuss sensitivity.

Solution

  1. 1
    S(10)=10log⁡10(10)=10×1=10S(10) = 10\log_{10}(10) = 10 \times 1 = 10.
  2. 2
    S(11)=10log⁡10(11)≈10×1.0414=10.414S(11) = 10\log_{10}(11) \approx 10 \times 1.0414 = 10.414.
  3. 3
    Change in SS: 0.4140.414 score points for an extra hour of study.
  4. 4
    The logarithm grows very slowly — sensitivity decreases as hh increases (diminishing returns).

Answer

ΔS≈0.414 points per extra hour at h=10\Delta S \approx 0.414 \text{ points per extra hour at } h=10
Logarithmic models have decreasing sensitivity: each extra unit of input produces smaller and smaller increases in output. This captures 'diminishing returns' mathematically.

About Sensitivity (Meta)

The degree to which a result or output changes in response to small changes in its inputs, parameters, or assumptions.

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