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Normal: percentage of defective washers

Short answer

Diameter X ~ N(0.502,0.005) cm. Acceptable range: 0.496–0.508, giving z=±1.20 with P(acceptable)=0.7698. Defective percentage = 1−0.7698 = 23.02%.

The mean inside diameter of a sample of 200 washers produced by a machine is 0.502 cm and the standard deviation is 0.005 cm. The purpose for which these washers are intended allows a maximum tolerance in the diameter of 0.496 to 0.508 cm, otherwise the washers are considered defective. Determine the percentage of defective washers produced by the machine. Assume the diameter is normally distributed.

Let X = inside diameter, X ~ N(μ = 0.502, σ = 0.005). A washer is non-defective when 0.496 ≤ X ≤ 0.508.

Standardise the limits

  • z₁ = (0.496 − 0.502)/0.005 = -1.20
  • z₂ = (0.508 − 0.502)/0.005 = 1.20

Probability of acceptable washers

P(0.496 ≤ X ≤ 0.508) = P(−1.20 ≤ Z ≤ 1.20) = 2 × 0.3849 = 0.7698

Defective washers

P(defective) = 1 − 0.7698 = 0.2302

(Equivalently P(Z < −1.2) + P(Z > 1.2) = 0.1151 + 0.1151 = 0.2302.)

Answer: about 23.02% of the washers are defective (roughly 46 out of 200).

✓ Answer checked by Claude Sonnet 5.5

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