Type I and Type II Errors Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
A medical test has α=0.05\alpha = 0.05 and β=0.20\beta = 0.20 (Power = 0.80). If the true disease rate is 5% in the population: (a) in 100 truly diseased patients, how many will be missed? (b) In 1000 truly healthy patients, how many will get false positives?

Solution

  1. 1
    Power = 1−β=0.801 - \beta = 0.80; so 80% of diseased patients are correctly detected
  2. 2
    (a) 100 diseased patients × β=0.20\beta = 0.20: 20 diseased patients missed (Type II errors)
  3. 3
    (b) α=0.05\alpha = 0.05: 5% of healthy patients falsely test positive; 1000 × 0.05 = 50 false positives (Type I errors)
  4. 4
    Summary: 80 true positives, 20 false negatives (missed); 50 false positives, 950 true negatives

Answer

(a) 20 missed diseased patients (Type II). (b) 50 false positives in 1000 healthy (Type I).
Concrete numbers make Type I/II errors tangible. Both types have real consequences: missed diseases (Type II) leave patients untreated; false positives (Type I) cause unnecessary treatment and anxiety. Optimal testing balances both, weighted by the relative costs of each error type.

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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