23-Ind-A5 Quality Planning, Control, and Assurance · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2013 — 98-Ind-A5 Quality Planning, Control and Assurance. Three-hour, closed-book exam; Casio or Sharp approved calculators only; one double-sided 8.5×11 aid sheet permitted; relevant statistical tables attached. Format: six questions, each worth 20 marks; any five constitute a complete paper, and only the first five appearing in the answer book are marked, so candidates effectively choose 5 of 6. All six are solved below for completeness.
Reference texts: Montgomery, Introduction to Statistical Quality Control (8th ed.) — control charts, process capability, acceptance sampling and quality management (the primary text for every part of this paper); ISO 9001:2015 — quality management systems and certification; MIL-STD-105E — sampling procedures and tables for inspection by attributes.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
A histogram summarizes a single continuous (or count) variable by grouping observed values into class intervals and plotting frequency — it answers "what does the distribution of this characteristic look like" (shape, centre, spread, skewness) and is built from measured/counted data on one variable. A Pareto chart instead ranks categories (defect types, complaint reasons, downtime causes) by frequency or cost, in descending bars, with a cumulative-percentage line overlaid — it answers "which few categories account for most of the problem," directly applying the Pareto (80/20) principle that a small number of causes typically drive most of the effect.
In process improvement, a Pareto chart is used first, to prioritize effort: e.g., a Pareto chart of solder-defect types on a PCB line might show that "cold joint" and "bridging" together account for 78% of all defects, telling the team exactly where to focus a root-cause investigation instead of spreading effort evenly across a dozen minor defect types. A histogram is then used to study the process behind the prioritized defect in more depth — e.g., a histogram of the solder-fillet height for cold-joint failures might reveal a distribution shifted well below the target fillet height, pointing toward a reflow-temperature or paste-volume root cause. Used together, the Pareto chart says where to look and the histogram helps explain why the process is behaving that way there.
Yes — a traditional Shewhart chart can be adapted to control a process with an inherent, expected trend (tool wear, chemical depletion, catalyst aging) as long as the trend itself is deterministic and predictable; the trick is to control charts around a moving target that follows the expected trend rather than around a fixed centre line, so the chart still flags abnormal deviation from the expected behaviour rather than flagging the expected trend itself as an out-of-control signal.
In discrete-parts industries, this is usually done with a regression (trend) control chart: a regression line (or a physically-derived wear model) is fit to the expected characteristic vs. time or vs. units produced (e.g., predicted bore diameter as tool wear accumulates), and the control limits are placed a fixed $\pm3\hat\sigma$ band around that fitted line rather than around a constant mean; a point is out of control when it departs from the predicted value by more than the band, e.g., signalling a broken or badly worn tool needing replacement earlier than the model predicts. In continuous-process industries, the same idea is applied as a chart on the residuals: an engineering/process model (or a time-series model such as an ARIMA/EWMA-based one-step-ahead predictor) forecasts the expected process output given the known drift mechanism (e.g., catalyst activity decay in a reactor), and the SPC chart is applied to the forecast error (observed $-$ predicted) rather than to the raw output — this is exactly the interaction between engineering process control and statistical process control: EPC/the trend model absorbs the predictable drift, and the residual chart looks for anything the model does not explain.
Both EWMA and CUSUM are "memory" charts, built to detect small, sustained shifts in the mean far faster than a memoryless $\bar X$ chart. The EWMA chart plots a geometrically weighted moving average $z_t=\lambda x_t+(1-\lambda)z_{t-1}$ ($0<\lambda\le1$), which smoothly discounts older observations; it is used, for example, to monitor a chemical process's key quality variable sample-by-sample where small drifts matter and $\lambda$ can be tuned (small $\lambda$ for very small shifts, $\lambda$ near 1 approaching the ordinary $\bar X$ chart). The CUSUM chart instead plots a running cumulative sum of deviations from target, $C_t^+=\max(0,\,x_t-(\mu_0+K)+C_{t-1}^+)$ and the mirrored $C_t^-$, which accumulates evidence linearly rather than geometrically; it is typically used when the objective is the fastest possible average detection of one specific reference shift size $\delta$ (e.g., monitoring fill weight where a shift of exactly $0.5\sigma$ must be caught quickly), because CUSUM's parameters ($K$, the reference value, and $H$, the decision interval) can be tuned precisely to that target shift and is provably close to optimal (minimum ARL) for detecting it.
The classical CUSUM is plotted and interpreted with a V-mask: a V-shaped template placed with its vertex a fixed lead distance $d$ ahead of the most recent point, and the process signals out of control the moment any earlier plotted point falls outside the two arms of the V. The V-mask's limitations are largely why most modern practice has abandoned it in favour of the algebraic tabular CUSUM: it is visually awkward and error-prone to apply consistently by hand (analysts must re-overlay the mask at every new point); the choice of lead distance $d$ and the half-angle $\theta$ is not intuitive and is easy to set inconsistently with the equivalent $(K,H)$ parameters; and, most importantly, a V-mask is inherently symmetric — it treats an upward and downward shift of the same magnitude identically, which is wrong whenever the practical cost or safety consequence of an upward shift differs from that of a downward shift (e.g., a fill-weight process where under-filling is a regulatory/customer-satisfaction problem but over-filling is "only" a material-cost problem).
Yes, a single CUSUM scheme can be designed with different positive and negative critical shifts: because the tabular CUSUM keeps two independent one-sided statistics, $C_t^+$ (with reference value $K^+=\mu_0+k^+\sigma$) for detecting upward shifts and $C_t^-$ (with $K^-=\mu_0-k^-\sigma$) for detecting downward shifts, the two reference constants $k^+$ and $k^-$ (and, if desired, the two decision intervals $H^+$ and $H^-$) can simply be set to different values — e.g., a small $k^-$ (tuned to detect a small downward shift quickly, for the under-fill risk) paired with a larger $k^+$ (tuned to a larger, more tolerable upward shift) — on a single chart displaying both cumulative statistics against their own, independently chosen limits.
