Question 6 of 7: Work Sampling — Purpose, Standard-Time Determination, and Statistical Accuracy

Question 6: Work Sampling — Purpose, Standard-Time Determination, and Statistical Accuracy (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) Basic Purpose and Applications of Work Sampling

Basic purpose. Work sampling estimates the proportion of time spent by a worker or machine in defined categories of activity — working, idle, on personal allowance, waiting for material — from a large number of instantaneous, randomly timed observations, rather than by continuously timing a single cycle with a stopwatch. Because each observation is a brief, momentary glance rather than sustained attendance, one observer can cover far more ground, and the operator is not subjected to the continuous, visible timing that provokes “watch consciousness.”

Applications. (1) measuring machine and labour utilization — the percentage of time equipment or personnel are actually productive; (2) determining the size of delay, personal and fatigue allowances empirically (Question 3(ii)) by directly observing how much time is actually spent in each category, rather than assuming a traditional flat percentage; (3) establishing time standards on long-cycle, irregular, group, or indirect-labour work where a stopwatch cycle time is difficult to even define (as demonstrated numerically in part (ii)); (4) analyzing how a worker or department splits time across multiple job types in mixed, non-routine work; and (5) auditing existing time standards for continued accuracy over a long period, catching informal method drift without continuous observation.

(ii) Standard Time via Work Sampling

Given.

Find. The standard time for the operation, in min./pc.

Approach. Convert the sampled idle percentage into the total time the operator was actually working, rate that working time to obtain a normal time per piece, then apply the total allowance.

1. Percentage of time working. $p_{\text{working}}=1-\dfrac{50}{150}=1-0.3333=\boxed{66.67\%}$ (the operator was idle for $50/150=33.3\%$ of the observed instants).
2. Total working time in the day. $T_{\text{working}}=480(0.6667)=\boxed{320\text{ min}}$.
3. Normal time per piece. Rate the working time and divide by the day’s output: $NT=\dfrac{T_{\text{working}}\times\text{rating}}{\text{pieces}}=\dfrac{320(1.50)}{250}=\dfrac{480}{250}=\boxed{1.92\text{ min/pc}}$.
4. Standard time. $ST=NT(1+A)=1.92(1.15)=\boxed{2.208\text{ min/pc}}$.

(iii) Relative and Absolute Accuracy of Operator Idle Time (Day 2, 99% Confidence)

Given. Day-2 sample: $n=300$ observations, operator idle in 75 of them; confidence level $=99\%$ ($z=2.576$).

Find. The absolute and relative accuracy of the estimated proportion of idle time.

Approach. Treat each observation as a Bernoulli trial (idle / not idle), so the sample proportion is binomially distributed and, for this sample size, well approximated by the normal distribution; compute the absolute accuracy from the standard confidence-interval half-width, then express it relative to the observed proportion.

1. Sample proportion idle. $p=\dfrac{75}{300}=\boxed{0.25\ (25\%)}$.
2. Absolute accuracy (half-width of the 99% confidence interval). $S_p=z\sqrt{\dfrac{p(1-p)}{n}}=2.576\sqrt{\dfrac{0.25(0.75)}{300}}=2.576\sqrt{0.000625}=2.576(0.025)=\boxed{0.0644\ (\pm6.44\text{ percentage points})}$.
3. Relative accuracy. $\dfrac{S_p}{p}=\dfrac{0.0644}{0.25}=\boxed{0.2576\ (\pm25.76\%\text{ of the estimate})}$.