Does a defect-free sample prove the whole shipment is good?

No. Finding zero defective units in a sample proves that none of the examined units failed the defined test. It does not prove that every unexamined unit is good. A shipment can pass a previously agreed sampling plan while still containing defects. The useful questions are what decision the plan supports, how the sample was selected and what risks the plan permits.

“All fifty passed” is a useful observation. “All thousand are perfect” is another claim entirely. Keep the first statement without quietly turning it into the second.

Choose the lot and decision before opening boxes

Identify the particular shipment, product version, lot boundary and unit being inspected. Define the requirement that makes a unit conforming or nonconforming. Then establish the sampling and disposition rules before seeing the results.

NIST's lot-acceptance guidance describes a plan as both a sampling scheme and decision rules. In a single-sample plan, n is the number examined and c is the permitted count of defective units before the lot is rejected.

Those two numbers are not a complete quality program. The actual product, consequences of failure, customer requirements, contract and applicable requirements determine what inspection and evidence are appropriate. A convenient sample size does not establish suitability for safety-critical use.

Also specify the unit being counted. One item with three problems is one defective item if the plan counts defective units. Counting three defect observations answers a different question and may require a different plan.

An original lot with a predefined zero-acceptance rule

Consider an entirely fictional delivery of 1,000 decorative wooden display blocks, all in one defined lot and specification. The particular test checks a specified dimension and cosmetic finish. It makes no finding about structural, electrical, food-contact or other safety requirements.

The supplied plan selects fifty distinct units by simple random sampling without replacement. Every possible group of fifty has the same chance of selection. The test is nondestructive and, for this mathematical illustration, perfectly identifies the defined nonconformity in an examined unit.

The previously fixed acceptance number is zero. If no sampled unit fails, the lot passes this supplied sampling rule. If one or more fail, it does not pass. These are invented teaching terms, not a recommended standard or a claim that fifty is enough for your product.

In the observed fictional branch, all fifty selected units pass. The honest result is zero failures among fifty inspected units, and acceptance under the stated rule. The other 950 units were not individually tested.

Because testing does not consume or alter the blocks, all 1,000 remain physical units in the lot. Acceptance does not create another fifty units or establish that the unexamined 950 independently passed.

Calculate how a bad lot could still pass

To understand the procedure, temporarily suppose a hypothetical lot of the same size contains exactly fifty defective units and 950 good units. This is a scenario for evaluating the plan, not a conclusion about the actual delivered lot.

For the zero-acceptance rule to pass that hypothetical lot, all fifty sampled units must come from the 950 good ones.

The exact probability is the number of ways to choose fifty from 950, divided by the number of ways to choose fifty from 1,000. Written with combination notation, that is C(950, 50) ÷ C(1,000, 50).

The result is approximately 7.20 percent. Even with fifty bad units in the shipment, this random-sampling procedure sometimes selects fifty good ones and accepts the lot.

One way to check the calculation is to multiply the chance of drawing a good unit at every step: 950/1,000, then 949/999, and so on through 901/951. Each draw removes a good unit from the hypothetical all-good selection path. This gives the same result as counting combinations.

This is a finite-lot calculation without replacement. Reusing 0.95 for every draw would assume the proportion stays unchanged after each selection. That simpler multiplication is not the exact model supplied here.

Compare two plans without changing the lot

In a separate alternative plan, select one hundred distinct units randomly from the same 1,000-unit hypothetical lot and keep the zero acceptance number. With fifty defective units present, the chance of selecting no defective unit becomes approximately 0.45 percent.

The larger sample is less likely to miss that particular level of nonconformity. It still does not make acceptance proof of zero defects.

Now examine a separate hypothetical lot containing only ten defective units and 990 good ones. A fifty-unit sample misses all ten about 59.73 percent of the time. A one-hundred-unit sample misses them about 34.69 percent of the time.

Scroll across to compare conditional acceptance probabilities. Each entry assumes the stated defective count is known for the hypothetical lot.

The same rule behaves differently at different underlying defect counts. That relationship is why one sample-size slogan cannot describe every risk. NIST calls the relationship between lot quality and acceptance probability an operating-characteristic curve.

Do not reverse the probability

The 7.20 percent answers: if this lot contains fifty defective units, how often would the stated fifty-unit procedure find none?

It does not answer: after finding none, what is the probability this actual lot contains fifty defective units? Nor does subtracting it from 100 produce a 92.80 percent probability that the delivered lot is good.

The calculation supplies no prior model of how many defects this supplier's lots contain. It also supplies no estimate of actual measurement error. You cannot recover those missing facts by changing the sentence around the same percentage.

A confidence bound, a supplier-process study and a probability model using prior information are different analyses. The survey-sample guide addresses estimating a population proportion under its stated design. This lot-disposition example is not that survey calculation applied to a box of goods.

Protect the selection and the test

Use a selection record that covers the whole defined lot. Checking the fifty easiest blocks to reach is not the supplied simple-random procedure. A supplier's specially prepared demonstration samples are not automatically a random draw from your received shipment either.

Retain selected unit references, specification version, test method, equipment information, inspector, observations and disposition. If units are inaccessible or replaced, record what happened and assess whether the design still applies.

Do not replace a failed selected item with another until you have fifty passing results. That changes the procedure and deletes the evidence the rule was meant to use.

The model assumes perfect classification. Real measurement needs an appropriate method and competent execution. A test that cannot detect the relevant failure will not acquire that ability merely because more units enter it.

The backup-supplier guide also distinguishes a passing qualification sample from production availability. Here the boundary is narrower still: the actual lot, actual selection and actual defined checks.

Fund the inspection you propose

For the fifty-unit fictional inspection, testing takes six minutes per item, totaling five hours. Setup takes one hour and complete documentation and disposition recording another hour. All necessary work therefore takes seven paid hours.

A qualified inspector has eight hours available without displacing accepted work. The complete cost is $35 for each hour actually used, so labor costs $245. Another $80 covers all test access, equipment use, handling, consumables and temporary holding for this bounded review.

The total is $325. A separate $400 inspection allocation leaves $75, and one of the eight available hours remains unused. No unpaid founder work or other inspection payment is required in this case.

This is only the review budget. Purchase, ordinary storage and operating commitments are protected elsewhere. It supplies no sales income, whole-company profit, future stock purchase or customer delivery promise.

For the separate one-hundred-unit proposal, testing takes ten hours, plus the same two supporting hours. Twelve hours at $35 cost $420; add the same complete $80 to reach $500.

Against the stated eight-hour availability and $400 fund, that larger proposal has a four-hour capacity gap and $100 cash gap. A lower theoretical miss probability does not make those missing resources available. Resolve them before calling the plan executable.

Decide what a failure means before it happens

If a sampled item fails the zero-acceptance rule, record the failure and follow the actual agreed disposition process. The fictional rule rejects the lot under that plan; it does not automatically establish a legal refund right or authorize shipping the rest anyway.

Any full sorting, replacement, repair, retesting or return needs its own permitted scope, time, money and acceptance evidence. None is secretly funded by the $75 left in this review budget. Hold unresolved goods under the appropriate process while the responsible parties determine the next action.

Likewise, do not keep taking fresh samples until one passes and then present only that result. A valid multiple-stage plan specifies how results combine. Improvised repeated chances can change the acceptance risk.

Use the inspection record to investigate what failed without inventing a cause. An out-of-specification dimension does not identify which machine, employee or supplier decision produced it.

Leave an honest acceptance record

Keep the lot boundary, complete selection rule, specification, sample size, acceptance number, actual observations and authorized disposition together. Separate the procedure's conditional risk calculations from findings about the delivered lot.

The strongest sentence may be the plainest: the fifty selected units passed these checks, and the lot met this agreed sampling rule. That is useful evidence with a visible boundary.