Two Technicians and a QA Failure Sample
A small refurbishing company has two technicians who prepare used tablets for resale. Technician A handles 60% of the tablets and has a 4% QA failure rate. Technician B handles 40% and has a 7% QA failure rate. Every tablet is tested once; failures are returned for rework. Last month a batch of 50 tablets was selected at random from all finished tablets, and the manager found that 4 failed QA. The manager says, 'B is almost twice as likely to produce a failure, so all rework blame should go to B, and we should move all work to A.' The operations lead is not convinced: A also produces failures, and the sample of 50 may not reflect the process. The company wants to know the overall failure rate, the probability that a failed tablet came from B, and whether the sample result is unusual enough to justify a staffing change.
1. Using the production shares and failure rates, find the overall QA failure rate and the probability that a failed tablet came from Technician B.
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Let F mean failed QA. P(F) = P(A)P(F|A) + P(B)P(F|B) = 0.60(0.04) + 0.40(0.07) = 0.024 + 0.028 = 0.052, or 5.2%. The failures from B are 2.8 percentage points of all tablets; failures from A are 2.4 percentage points. By Bayes, P(B|F) = 0.028 / 0.052 ≈ 0.5385, about 53.8%. So B contributes slightly more than half of failures, not almost all. A contributes 0.024 / 0.052 ≈ 46.2%, nearly half.
2. If the process rates hold, what is the expected number of failures in a sample of 50 tablets? Estimate the probability of seeing 4 or more failures and interpret whether 4 failures is strong evidence that something changed.
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Expected failures = 50 × 0.052 = 2.6. Treating failures as binomial with n = 50 and p = 0.052, P(X ≥ 4) is about 0.26. That means even if the process is unchanged, roughly one sample in four would show 4 or more failures. Four failures is above the mean of 2.6, but it is not a rare event. The sample alone is weak evidence of a real shift. A larger sample, or a run chart over several months, would be needed before concluding that the failure rate has changed.
3. Evaluate the manager's conclusion that all blame should go to B and that all work should move to A. Use the computed shares and the sample result.
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B's per-tablet failure rate is higher (7% vs 4%), but B handles only 40% of tablets, so B accounts for about 53.8% of failures while A accounts for about 46.2%. The statement that B deserves all blame is therefore false. Moving all work to A would lower the expected overall failure rate from 5.2% to 4.0% only if A's failure rate and capacity stayed unchanged, a 1.2 percentage point improvement. However, the observed 4 failures in 50 is consistent with normal variation (P ≥ 4 is about 0.26), so it does not establish that the process changed or that B is the sole problem. The better response is to investigate causes at both stations and gather more data before reassigning all work.