Question 3 of 7: Rating and Allowances in Stopwatch Time Study, and Required Sample Size

Question 3: Rating and Allowances in Stopwatch Time Study, and Required Sample Size (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.

(i) Why Rating and Allowances Are Critical and Controversial, and How to Alleviate the Problems

Performance rating and allowances are critical because they are the two adjustments that convert a raw stopwatch reading — specific to one operator, on one day, under one set of conditions — into a standard intended to be fair and achievable for any qualified worker under normal conditions indefinitely. Without a correct rating, a standard timed on a fast operator becomes unattainable by an average worker, and one timed on a slow operator overpays for ordinary performance; without correct allowances, even a perfectly rated normal time ignores the unavoidable non-productive time (personal needs, unavoidable delays, fatigue) that a real shift always contains. Both are controversial precisely because both ultimately rest on a trained analyst's judgement rather than on a directly measured physical quantity: two analysts observing the identical performance can, in good faith, assign different ratings, and the "correct" size of a fatigue or delay allowance for a given job is rarely backed by a plant-specific measurement. Because pay (under incentive systems) or output expectations are riding on the resulting standard, small differences in judgement translate directly into disputes between labour and management over whether a standard is fair.

Industry alleviates these problems through several complementary approaches: (1) rating training and certification, using synchronized rating films or videos of known, independently verified paces so analysts practise until their individual ratings converge within an accepted tolerance, with periodic re-certification; (2) consensus/group rating, averaging two or more trained analysts' independent ratings of the same observation to dilute any one analyst's bias; (3) adopting predetermined motion-time systems (MTM, MOST — Question 5) that replace subjective rating altogether with published, motion-level time values derived once from a large independent film study; (4) determining allowances empirically via work sampling (Question 6) — observing actual personal, delay and fatigue time directly across many random instants — rather than applying a traditional blanket percentage; and (5) involving worker/union representatives in setting and auditing rating and allowance practice, which improves the perceived fairness of the resulting standards even where the underlying judgement cannot be made perfectly objective.

(ii)(a) Range of Elemental Time Values and Percentage Accuracy at 95% Confidence

Given.

Find. The 95% confidence range for the true mean elemental time, and the corresponding percentage accuracy of the sample mean.

Approach. Compute the standard error of the mean from the sample, apply the $z$-multiplier for 95% confidence to get the margin of error, add/subtract it from the sample mean for the range, and express the margin as a percentage of the mean for the accuracy level.

1. Standard error of the mean. $SE=\dfrac{s}{\sqrt{n}}=\dfrac{0.06}{\sqrt{25}}=\dfrac{0.06}{5}=\boxed{0.012\text{ min}}$.
2. Margin of error at 95% confidence. $E=z_{0.025}\times SE=1.96(0.012)=\boxed{0.02352\text{ min}}$.
3. Range of elemental time values. $\bar{X}\pm E=0.20\pm0.02352$, i.e. $\boxed{0.17648\text{ to }0.22352\text{ min}}$.
4. Percentage accuracy. $\text{Accuracy}=\dfrac{E}{\bar{X}}\times100\%=\dfrac{0.02352}{0.20}\times100\%=\boxed{11.76\%}$.

(ii)(b) Required Number of Observations for 10% Accuracy at 95% Confidence

Given. Same sample as part (a) ($n=25$, $\bar{X}=0.20$ min, $s=0.06$ min), desired accuracy $k=10\%=0.10$ of the mean, 95% confidence ($z=1.96$).

Find. The total number of observations $N'$ required so the sample mean is within 10% of the true mean at 95% confidence, and the number of additional readings needed beyond the 25 already taken.

Approach. Use the time-study sample-size formula $N'=\left(\dfrac{zs}{k\bar{X}}\right)^2$, which rearranges the accuracy formula of part (a) to solve for the sample size that would deliver a target accuracy $k$, then round up to the next whole reading and subtract the 25 already collected.

1. Required sample size. $N'=\left(\dfrac{zs}{k\bar{X}}\right)^2=\left(\dfrac{1.96(0.06)}{0.10(0.20)}\right)^2=\left(\dfrac{0.1176}{0.02}\right)^2=(5.88)^2=34.57$.
2. Round up to a whole number of readings. $N'=\boxed{35\text{ observations}}$ (a fractional reading is not possible, so the required count always rounds up).
3. Additional readings needed. $35-25=\boxed{10\text{ additional readings}}$, beyond the 25 already taken.