The Quality-Control Sampling Decision
A factory produces circuit boards with a historical defect rate of 4%. A new production line is installed, and an inspector samples 150 boards, finding 12 defective. Management claims the new line is worse because 12/150 = 8% exceeds 4%. A statistician notes that sampling variation could easily produce 12 defects even if the true rate is still 4%. The inspector must decide whether to halt production, using a normal approximation to the binomial distribution with mean np and standard deviation √(np(1−p)). A second sample of 150 boards from the same line yields 5 defective. The team must reconcile the two samples and decide what to report.
1. Using the normal approximation, assess how unusual a result of 12 defects would be if the true defect rate were 4%.
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With n = 150 and p = 0.04: mean = np = 6, variance = np(1−p) = 6(0.96) = 5.76, standard deviation = √5.76 = 2.4.
Standardize with continuity correction: z = (12 − 0.5 − 6)/2.4 = 5.5/2.4 ≈ 2.29.
The probability of 12 or more defects is roughly P(Z ≥ 2.29) ≈ 0.011. Without continuity correction, z = (12 − 6)/2.4 = 2.5, giving ≈ 0.006.
Either way, a result this extreme is unlikely (about 1% or less) under the 4% assumption. The judgement call is that "unlikely" is not "impossible," and a single sample at the 1% level is suggestive but not conclusive, especially when multiple lines or repeated testing could produce occasional extremes by chance.
2. Combine the two samples and decide what the evidence supports.
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Combined: 12 + 5 = 17 defective out of 300 boards, giving 17/300 ≈ 5.67%.
Under the null p = 0.04: mean = 300(0.04) = 12, variance = 12(0.96) = 11.52, sd = √11.52 ≈ 3.394.
z = (17 − 0.5 − 12)/3.394 = 4.5/3.394 ≈ 1.33, giving P(Z ≥ 1.33) ≈ 0.092.
So the combined evidence is not significant at the 5% level. The first sample looked alarming, but pooling with the second pulls the estimate toward 4%. The judgement is that the first sample was likely an unlucky fluctuation, and halting production on it alone would be premature. The team should report the combined rate of about 5.7% with its uncertainty and recommend continued monitoring rather than an immediate shutdown, since the pooled result is consistent with the historical rate at conventional significance levels.