Question 4 of 6: Attribute Charts, the Case for a Positive $c$-Chart LCL, and a $u$-Chart for Paper-Mill Imperfections

Question 4: Attribute Charts, the Case for a Positive $c$-Chart LCL, and a $u$-Chart for Paper-Mill Imperfections (20 marks)

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) Points on the limits; $p$/$np$/$c$/$u$ differences; demerit charts

Attribute statistics ($p$, $np$, $c$, $u$) are inherently discrete — a count or proportion can only take specific values — so the exact probability associated with a plotted point that lands exactly on the computed limit is generally not negligible and the true, achievable probability of exceeding the limit can differ noticeably from the nominal $3\sigma$ target; a point that lands precisely on the boundary is therefore treated as investigation-worthy, since with a coarse, discrete statistic that boundary point is often effectively equivalent to "beyond" it. A variable chart's statistic is continuous, so the probability of landing exactly on a continuous limit is zero — a point that close is genuinely uninformative and does not itself warrant investigation.

The four attribute charts differ in what they count and whether the sample size is fixed. The $p$-chart plots the fraction nonconforming and allows the sample size to vary from sample to sample. The $np$-chart plots the raw count of nonconforming units and requires a constant sample size — it carries the same information as a $p$-chart with fixed $n$, but is simpler for shop-floor operators to plot as a whole number. The $c$-chart plots the count of nonconformities (defects, of which a single unit can have more than one) found in one constant-size inspection unit — e.g. one car body, one roll of paper. The $u$-chart plots nonconformities per unit and allows the inspection-unit size (the area of opportunity) to vary — it is to $c$ what $p$ is to $np$. A demerit chart is used when different types of nonconformity are of different severity: each defect class (e.g. critical/major/minor) is assigned a demerit weight, and the single plotted statistic is the total weighted demerit score per unit or per inspection unit — giving the chart far more sensitivity to serious defects than a plain $c$- or $u$-chart, which would count a critical defect and a cosmetic scratch identically.

(b) Why a positive $c$-chart LCL is preferred, and how to get one

Quality engineers prefer $LCL_c=\bar c-3\sqrt{\bar c}$ to come out strictly positive (rather than clamp at 0) because only a positive LCL lets the chart signal on the low side — an unusually small nonconformity count is then a statistically meaningful event worth investigating, and if it traces to a genuine improvement (a better raw-material batch, a process tweak, an operator technique), that improvement can be identified and made permanent. If $LCL_c$ is clamped at 0 — which happens automatically whenever $\bar c<9$, since $\bar c-3\sqrt{\bar c}>0$ requires $\bar c>9$ — the chart can never register a low-side signal, and genuine process improvements go undetected and un-institutionalized. The practical way to achieve $\bar c>9$ is to increase the size of the inspection unit — inspect more items, more area, or a longer length per sample (or, equivalently, switch to a $u$-chart with a larger $n$ per point) — so the expected nonconformity count per inspection unit rises comfortably above 9.

(c) $u$-chart for paper-mill imperfections, 20 days, variable rolls/day

Given. The number of rolls produced (the inspection-unit count) and the total imperfections found vary day to day over 20 days:

Find. An appropriate 3-sigma chart for these data, revised if necessary, and $\lambda$, the estimated expected number of nonconformities per roll.

Approach. Because the "inspection unit" (a roll of paper) is fixed but the number of inspection units per day varies (18, 20, or 22 rolls), a $u$-chart — nonconformities per roll, with day-specific 3-sigma limits — is the appropriate chart (a $c$-chart would require a constant number of rolls per day, which is not the case here).

1. Pooled nonconformities per roll. $\sum(\text{rolls})=400$, $\sum(\text{imperfections})=285$, so $\bar u=\dfrac{285}{400}=\boxed{0.7125}$ imperfections/roll.
2. Control limits by daily roll count. $UCL_i=\bar u+3\sqrt{\bar u/n_i}$, $LCL_i=\max(0,\bar u-3\sqrt{\bar u/n_i})$. For $n_i=18$: $UCL=0.7125+3\sqrt{0.7125/18}=\boxed{1.309}$, $LCL=\boxed{0.116}$. For $n_i=20$: $UCL=\boxed{1.279}$, $LCL=\boxed{0.146}$. For $n_i=22$: $UCL=\boxed{1.252}$, $LCL=\boxed{0.173}$.
3. Check every day. Day 6 ($22$ rolls, $28$ imperfections, $u=28/22=1.2727$) exceeds its $UCL=1.2524$ — every other day's $u_i$ falls inside its own limits. Day 6 is removed as an assignable-cause point and the chart is revised.
4. Revised pooled rate. Excluding day 6: $\sum(\text{rolls})=378$, $\sum(\text{imperfections})=257$, so $\bar u_{rev}=\dfrac{257}{378}=\boxed{0.6799}$. Re-checking all 19 remaining days against limits recomputed from $\bar u_{rev}$ (e.g. at $n_i=22$, $UCL_{rev}=0.6799+3\sqrt{0.6799/22}=1.208$) leaves every remaining day inside its own limit — no further revision is needed.
5. Estimate $\lambda$. The expected number of nonconformities per roll is the revised in-control rate itself, $\lambda=\bar u_{rev}=\boxed{0.680}$ imperfections/roll.