Confidence intervals, sample size and bias in work sampling

Why it matters

The 30% waiting result in work sampling does not come from observing every eligible moment in scope one by one. It is an estimate calculated from selected moments. If another random sample were collected with the same design, the proportion might change slightly. The confidence interval quantifies this sampling variability. A narrow interval requires an adequate number of observations that are suitably spread out. But a narrow interval does not prove that the selection was right. A large number of records collected only from calm hours can produce a result that looks precise about the whole shift yet is biased. The universe, the selection and the independence conditions the calculation rests on must be as explicit as the number.

The approximate calculation used in this lesson splits each valid observation into two outcomes: waiting or no waiting. It assumes that the selected moments represent the same target universe and that the observations can be treated as independent. Even when there are more classes, the proportion of a given class can be examined with this binary split. Very close records inside a single long wait, however, should be questioned for independence. It is not enough that every row has a different identifier. When the same cause affects a group of records together, the size visible in the table and the amount of information carried are different things. Seeing the same class within the same event does not by itself prove statistical dependence; an independent random time draw has to be told apart from a record pattern that is clustered around an event or taken at fixed intervals.

The error target is set by which difference the decision has to resolve. Around 30%, ±5 percentage points means a planning width between 25% and 35%. 5% of the proportion in relative terms is 0.30 × 0.05 = 0.015, that is, 1.5 percentage points. The two targets are not the same. Entering the margin of error into the formula as 5 is also wrong; on the proportion scale 0.05 is used. The required number of observations is inversely proportional to the square of the error. A narrower target therefore increases the observation load quickly. The error target is not loosened just to get an easy result without changing the research question.

A 95% confidence statement is a long-run coverage property of the interval-building method that was applied, under similar conditions. It does not mean that the true proportion will stay inside this interval every day, or that 95% of the workers will behave the same way. The calculated interval is reported with the specific sample and assumptions. For the approximate normal calculation, besides independence, both classes need enough observations; in this course, n×p and n×(1−p) being at least 10 is used as a preliminary check. This threshold is not an exact coverage guarantee. With a small sample or extreme proportions, a suitable proportion interval, for example a score interval or an exact binomial interval, is considered. In such cases, forcibly clipping the interval to between zero and one does not make the problem solved. The interval method and the sampling design are assessed separately. The aim of this lesson is not to apply a single formula to every data structure but to recognize the condition under which it can be used.

Formula