During a quarterly quality-control audit, an engineer randomly selects 15 memory modules from a warehouse of approximately 5 000 units without replacement and records how many are defective. She plans to model the count of defectives with a Binomial(15, p) distribution to build a confidence interval for the unknown defect rate. Which fundamental assumption required by the binomial model is most likely violated by this sampling design and, if ignored, will typically over-state the sampling variance?
Both np and n(1 − p) must be at least 5 to justify a normal approximation.
Each trial outcome is independent of all other trials.
Every module can be classified into exactly two mutually exclusive states (defective or non-defective).
The total number of trials is fixed in advance at 15.
The binomial distribution assumes that every trial is independent; the outcome of one trial must not affect the probability of success on any other trial. Drawing items without replacement introduces negative dependence between draws-once a defective module is selected, the chance of selecting another defective on the next draw decreases. The correct model in this setting is the hypergeometric distribution, whose variance equals np(1 − p) multiplied by the finite-population correction (N − n)/(N − 1); this factor is less than 1, so using the binomial variance over-states the spread. The other listed conditions (binary outcome, fixed sample size) are satisfied, and the rule-of-thumb about np and n(1 − p) ≥ 5 is a guideline for normal approximation, not a core binomial assumption.
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