Uncertainty Math Example 4

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Example 4

hard
Two studies estimate the same parameter: Study A: ฮธ^=10ยฑ2\hat{\theta} = 10 \pm 2; Study B: ฮธ^=12ยฑ1\hat{\theta} = 12 \pm 1. Are these results consistent or contradictory? How would you combine them?

Solution

  1. 1
    Study A interval: [8,12][8, 12]; Study B interval: [11,13][11, 13]
  2. 2
    Overlap: both intervals include [11,12][11, 12] โ€” the results are consistent (not contradictory)
  3. 3
    Combining: weight each estimate by inverse variance; wA=1/4w_A = 1/4, wB=1/1=1w_B = 1/1 = 1
  4. 4
    Weighted mean: ฮธ^combined=(1/4)(10)+(1)(12)1/4+1=2.5+121.25=14.51.25=11.6\hat{\theta}_{\text{combined}} = \frac{(1/4)(10) + (1)(12)}{1/4 + 1} = \frac{2.5 + 12}{1.25} = \frac{14.5}{1.25} = 11.6

Answer

Results overlap at [11,12][11,12] โ€” consistent. Combined estimate โ‰ˆ 11.6 (weighted toward more precise Study B).
Consistency is checked by interval overlap. Combining estimates from multiple studies (meta-analysis) uses inverse-variance weighting: more precise studies (smaller uncertainty) get higher weight. This is the basis of meta-analytic methods in research synthesis.

About Uncertainty

Uncertainty is the state of having incomplete or imperfect information about a quantity, outcome, or process, making precise prediction impossible.

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