The Defective Batch
A factory produces electronic components. Each component is tested for defects. The test is not perfect: if a component is defective, the test correctly identifies it as defective 95% of the time (sensitivity). If a component is not defective, the test correctly identifies it as not defective 90% of the time (specificity). The factory's production process has a defect rate of 2%. A component is selected at random and tested. The test result is positive (defective). The factory manager wants to know the probability that the component is actually defective. Additionally, the factory is considering improving the test to have a specificity of 95% while keeping sensitivity at 95%. The cost of improvement is $50,000. If a defective component is shipped, it causes $100 in warranty costs. If a non-defective component is scrapped due to a false positive, it costs $50 in lost profit. Currently, the factory produces 100,000 components per year.
1. Calculate the probability that a component is actually defective given a positive test result (positive predictive value).
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Let D = defective, P = positive test. We know: P(D) = 0.02, P(not D) = 0.98. Sensitivity P(P|D) = 0.95. Specificity P(not P|not D) = 0.90, so false positive rate P(P|not D) = 0.10. By Bayes' theorem: P(D|P) = P(P|D)P(D) / [P(P|D)P(D) + P(P|not D)P(not D)] = (0.950.02) / (0.950.02 + 0.10*0.98) = 0.019 / (0.019 + 0.098) = 0.019 / 0.117 ≈ 0.1624, or about 16.24%.
2. Calculate the expected annual cost of warranty costs for shipped defective components and scrapping costs for false positives under the current test. Then, calculate the expected annual cost if the test is improved to specificity 95%. Should the factory invest in the improved test?
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Current test: For 100,000 components, expected defective = 2% = 2,000. Expected non-defective = 98,000. True positives (defective and test positive) = 2000 * 0.95 = 1,900. These are detected and not shipped? Wait, the problem says: if a defective component is shipped, it causes $100 warranty cost. So defective components that are not detected (false negatives) are shipped. False negatives = defective and test negative = 2000 * 0.05 = 100. So 100 defective shipped, warranty cost = 100 * $100 = $10,000. False positives (non-defective but test positive) = 98,000 * 0.10 = 9,800. These are scrapped, cost = 9,800 * $50 = $490,000. Total cost = $10,000 + $490,000 = $500,000.
Improved test: specificity 95%, so false positive rate = 5%. Sensitivity still 95%. False negatives still 100 (same sensitivity). Warranty cost = $10,000. False positives = 98,000 * 0.05 = 4,900. Scrap cost = 4,900 * $50 = $245,000. Total cost = $10,000 + $245,000 = $255,000. Savings = $500,000 - $255,000 = $245,000 per year. Cost of improvement = $50,000 one-time. So payback is less than 3 months. Yes, invest.