Errors and Power: Detecting a Difference Without Claiming Certainty — Solutions and Rubrics

Type I: the population mean is really 18.000 kg, but the sample triggers rejection and apparent evidence of a different mean. Type II: the population mean differs from 18.000 kg, but the rule fails to reject, missing the shift. Power at a specified shifted mean is the probability of correctly rejecting the false target null. A small p-value can cause the Type I decision even though the alternative is false; a large p-value can cause the Type II decision even though the alternative is true.

SE=0.010/√5=0.00447214 kg. The half-width is 1.96SE=0.00876539 kg; failure region is [17.9912346,18.0087654]. Under μ=18.000, α=2[1−Φ(1.96)]≈0.0499958. At μ=18.012, β=P(lower≤x̄≤upper)=Φ(-0.7232816)−Φ(-4.6432816)≈0.234751741. Power=1−β≈0.765248259. The interval probability is a missed shift under the specified false null; the complement is correct detection.

If the true population mean is 18.012 kg, about 23.475% of five-packet checks fail to flag the shift and about 76.525% flag it. Both assume the target null is false at this alternative. Failure to reject does not establish a true target; here it misses a false target. Alpha instead concerns rejection when the target mean is true.

Type I: reject a shipment whose true p is at the baseline 0.02; probability α=0.01. Type II at p=0.035: fail to flag a shipment with that harmful population contamination; β=1−0.76=0.24. Detection probability is power=0.76 at p=0.035. At a different true p, the distance from the null and therefore β/power generally differ; α is specified under the null, not as the Type II probability.

Missed contamination can harm customers, so Type II consequences may be more serious here, though false destruction also has costs. A different setting could prioritize false alarms; compare consequences before the study. Lowering α makes false rejection at the baseline less likely but makes rejection harder, tending to increase β and reduce power for the fixed harmful alternative. Increasing n reduces sampling variation and tends to reduce β/increase power at the same α; sample-size planning can address missed harm without simply raising the false-alarm rate. Type I seriousness informs α; Type II seriousness informs n.

Baseline power=0.32821263, β=0.67178737. A: power=0.52081624, β=0.47918376; larger n reduces SE. B: power=0.44241307, β=0.55758693; less variability gives smaller SE at the same n. C: power=0.71060260, β=0.28939740; true value farther from the null is easier to distinguish. D: power=0.46750149, β=0.53249851; higher alpha lowers the rejection threshold. Each β=1−power. Other relevant stated settings are fixed in each comparison. Larger n commonly acts through reduced SE, so n and SE are connected mechanisms rather than independent physical switches.

Power is a probability of rejection assuming a specified false null, not evidence that any current null is false. No test guarantees a correct individual decision. Increasing α trades more detection for more Type I error under the null. If false alarms are expensive, larger n or lower measurement variability may improve power while retaining α=0.05; feasibility and cost still matter. The numerical model alone does not select a universal best policy.