FE Industrial & Systems Practice Question Solution included after investigation
CASE E-087 · FE

The Sample Mean That Crossed 53, Even Though 53.5 Looked Safe

The individual-process spread looked wide enough to contain 53.5, but an x̄ chart uses the standard error of the sample mean. The upper control limit was 53.0.

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Case E-087 engineering failure visual
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Follow the investigation process used in the field — in five guided steps.

1

Inspect

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2

Commit

Choose your hypothesis.

3

Prove

Run calculations and test your idea.

4

Fix

Select and validate a safe correction.

5

Sign Off

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Case Brief

SAME PRINCIPLES. HIGHER STANDARDS.
PROBLEM STATEMENT

A stable process has historical mean μ=50.0 mm and known standard deviation σ=2.0 mm. Quality engineers monitor subgroup means using samples of n=4 and three-sigma x-bar limits. A new subgroup has mean 53.5 mm. A note compares 53.5 with μ±3σ=44 to 56 mm and labels the subgroup in control. Determine the correct x-bar control limits and status of the subgroup.

Quality Industrial Systems Design Verification Root Cause Analysis Safe Correction
E-087
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CASE CLOSED · ENGINEERING VERDICT

What actually failed — and what should change.

What failed

Individual-observation limits were used for an x-bar chart.

What was missed

Sample means have lower variability than individual measurements.

Corrected model

Use μ±3σ/√n for the stated x-bar chart.

QUICK ANSWER

For μ=50 mm, σ=2 mm, and n=4, SE=2/√4=1 mm. Three-sigma x-bar limits are 50±3(1), or 47 to 53 mm. A subgroup mean of 53.5 mm is therefore out of control.

ENGINEERING TAKEAWAY

For an x-bar chart with known σ and subgroup size n, use UCL/LCL=μ±3σ/√n.

CASE QUESTIONS

Questions this case should settle.

What standard deviation is used for an x-bar chart with known sigma?

The standard error σ/√n.

What happens to x-bar limits as subgroup size increases?

They become narrower because σ/√n decreases.

NEXT FE PRACTICE QUESTION

The 5% Defect Lot That Was Accepted 73.6% of the Time, Not 95%

Quality · 5–10 min

Continue Practice →
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  • Strengthen exam-ready thinking through realistic scenarios
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