Quality Engineering and Process Improvement Exercises

These exercises practise quality engineering and process improvement decisions using measurement evidence, process capability, defect data, risk ranking, corrective action, and validation statistics. The purpose is not only to calculate a quality metric. The purpose is to decide whether a product, process, supplier lot, or corrective action is credible enough to release or whether more control is required.

Assume simplified normal-distribution and sampling assumptions unless an exercise states otherwise. Real quality decisions should also check measurement-system bias, calibration, lot traceability, operator method, environmental effects, supplier changes, product mix, severity of escape, and regulatory or customer requirements.

How to Use These Exercises

For each quality calculation, define:

1. the requirement or risk being controlled;
2. the measurement method and uncertainty;
3. the decision rule for accept, reject, hold, release, or escalate;
4. the evidence needed to close corrective action;
5. the operational consequence of a false accept or false reject.

The common mistake is treating quality as inspection after the fact. A useful quality metric changes a control plan, sampling plan, supplier requirement, design tolerance, validation plan, or corrective action.

For each result, state whether it supports release, hold, rejection, process adjustment, supplier escalation, CAPA closeout, validation redesign, or measurement-system improvement. Quality calculations should make the decision rule and consequence of error explicit.

Exercise 1: Metrology Guard Band

A precision bore has specification:

The expanded measurement uncertainty is:

Use a conservative guard band by subtracting uncertainty from both specification limits. Find the guarded acceptance interval.

Solution

Specification limits:

Guarded lower limit:

Guarded upper limit:

The guarded acceptance interval is:

Engineering Comment

The guard band reduces false acceptance risk but may increase false rejects. The correct rule depends on measurement capability, process capability, cost of rejection, and the consequence of releasing a nonconforming bore.

Exercise 2: Process Capability and Centering

A shaft diameter has:

The process mean is:

and standard deviation is:

Calculate C_p and C_{pk}.

Solution

Process potential:

Upper-side capability:

Lower-side capability:

Therefore:

Engineering Comment

The process spread could be acceptable, but the process is off-center toward the upper limit. Improvement should first recenter the process before investing only in variation reduction.

Exercise 3: Z-Score for an Outlying Measurement

A process characteristic has historical mean:

and standard deviation:

A new measurement is:

Calculate its z-score.

Solution

The z-score is:

Engineering Comment

The point is 2.5 standard deviations below the historical mean. That may justify investigation, especially if the characteristic is critical or if several points shift in the same direction. The decision should consider measurement error and whether the historical distribution is stable.

Exercise 4: Gage R&R as Percent of Tolerance

A measurement-system study estimates combined repeatability and reproducibility standard deviation:

The product tolerance width is:

Estimate percent tolerance consumed by measurement variation using 6\sigma_{GRR}.

Solution

Measurement spread:

Percent of tolerance:

Engineering Comment

A 36 percent measurement-system burden is usually too high for confident process control. The team should review fixture design, resolution, operator method, part location, environmental effects, and calibration before using the data to accept or reject product.

Exercise 5: Sampling Decision for a Supplier Lot

A receiving inspection plan samples:

from a supplier lot. The acceptance number is:

meaning the lot is accepted if the sample has 0 or 1 defect. The sample contains:

State the lot decision.

Solution

The observed defect count is:

The acceptance criterion is:

Since:

the lot fails the sampling plan and should be rejected, held, or escalated according to the quality agreement.

Engineering Comment

The sampling plan is a decision rule, not proof that every unit is bad. The next action should consider defect severity, containment, supplier history, production need, customer risk, and whether 100 percent sorting is technically capable.

Exercise 6: FMEA RPN and Residual Risk

A process failure mode has:

An interlock reduces occurrence to 2 and detection ranking to 3. Calculate initial and revised RPN.

Solution

Initial RPN:

Revised RPN:

Reduction:

Engineering Comment

The RPN falls strongly, but severity remains 8. If the effect is safety-critical, the residual risk may still require independent protection, validation testing, management approval, or design change.

Exercise 7: CAPA Recurrence Evidence

A defect rate before corrective action was:

After corrective action, three normal-production lots show:

Lot	Defects	Units inspected
1	6	800
2	4	900
3	5	1100

The closeout criterion is fewer than 6 defects per 1000 units. Check whether the criterion is met.

Solution

Total defects:

Total inspected:

Post-action defect rate:

Since:

the recurrence criterion is met.

Reduction from baseline:

Engineering Comment

The result supports closeout only if the lots represent normal production, not special sorting. CAPA evidence should also show that the root cause was addressed and the control plan was updated.

Exercise 8: Pareto Defect Priorities

A defect review finds the following monthly counts:

Defect type	Count
Burr	44
Scratch	21
Missing label	15
Leak	12
Wrong torque	8

Find the cumulative percentage for the top two and top three defect types.

Solution

Total defects:

Top two:

so:

Top three:

so:

Engineering Comment

The top three categories account for 80 percent of defects. That is a useful prioritization signal, but severity still matters. A low-count leak or wrong-torque defect may deserve action before a high-count cosmetic defect if the consequence is higher.

Exercise 9: Validation Sample Size and the Rule of Three

A validation run tests:

with zero observed failures. Use the rule of three to estimate the approximate one-sided 95 percent upper bound on failure probability:

Compare with a requirement to demonstrate failure probability below 1 percent.

Solution

Upper bound:

The zero-failure run does not demonstrate failure probability below 1 percent at this confidence level.

Engineering Comment

Zero observed failures is not the same as zero risk. If the requirement is below 1 percent, a larger sample, stronger prior evidence, accelerated testing, or a different validation argument is needed.

Exercise 10: Improvement Benefit from Rework Reduction

A process produces:

The rework rate falls from:

to:

after a fixture improvement. Rework costs:

and technician time is valued at:

Estimate monthly labor cost avoided.

Solution

Reworked units avoided:

Rework time avoided:

Labor cost avoided:

Engineering Comment

This is only the direct labor saving. The real benefit may also include shorter lead time, lower WIP, fewer escapes, less operator frustration, and more bottleneck capacity. The control plan should verify that the rework reduction persists.

Review Checklist

A strong quality engineering solution should check:

• whether the requirement, tolerance, decision threshold, and consequence of escape are explicit;
• whether measurement uncertainty, bias, calibration, resolution, repeatability, reproducibility, and operator method are controlled;
• whether guard bands balance false acceptance against false rejection for the actual risk;
• whether capability metrics are based on a stable process and enough representative data;
• whether sampling decisions account for defect severity, supplier history, containment, and sorting capability;
• whether FMEA reductions leave high-severity residual risk requiring design control or approval;
• whether CAPA evidence proves root-cause correction under normal production conditions;
• whether validation sample size and confidence match the claimed failure probability.

The result should change a control plan, validation plan, supplier action, process setting, or corrective-action record.