Type I and Type II Errors Math Example 4

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Example 4

hard
A factory quality test accepts shipments if sample defect rate is below 5%. H0H_0: defect rate ≤ 5% (accept). Type I: reject good shipment. Type II: accept bad shipment. Which error is more costly for the factory, and how should this affect choice of α\alpha?

Solution

  1. 1
    Type I (reject good shipment): wastes a valid shipment — economic loss but no safety risk
  2. 2
    Type II (accept bad shipment): bad products reach customers — potential safety issues, recalls, reputation damage, legal liability
  3. 3
    Type II is likely more costly in this scenario
  4. 4
    Implication: use a larger α\alpha (e.g., 0.10) to make it easier to reject shipments, reducing Type II errors (accepting bad batches)

Answer

Type II (accepting bad shipment) is more costly → use larger α\alpha to detect problems more aggressively.
The choice of α should reflect the relative costs of each error type. When Type II errors are most costly (e.g., safety-critical manufacturing), use a larger α to prioritize sensitivity. When Type I errors are most costly (e.g., wrongful conviction), use a smaller α.

About Type I and Type II Errors

Type I error (α\alpha): rejecting H0H_0 when it is actually true (false positive). Type II error (β\beta): failing to reject H0H_0 when it is actually false (false negative).

Learn more about Type I and Type II Errors →

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