Sampling

2.1 Choosing the proper sample size An engineer responsible for resist processing had to address the following problem. His resist supplier provided materials with batch-to-batch photospeed controlled to ± 3%. For his process, this corresponded to linewidth changes of ± 8 nm about the target value of 250 nm. The engineer wanted to determine the photospeed of a new batch of resist. This information would be used to adjust exposure doses in order to continue to pattern production wafers with linewidths at their target values. The engineer wanted to expose the minimum number of wafers in order to guarantee that the linewidths would change, on average, by no more than ± 1.5 nm after the facility changed to the new batch of resist. Because the process had been running production volumes for several months, the engineer knew that the process with the old batch of resist was well centered on the target, that is, the process mean The engineer also knew the (short term) wafer-to-wafer variation very well: Keeping the exposure dose fixed, it was reasonable to assume that σ would be unchanged with the new batch of resist, only that μ would change. Because of the wafer-to-wafer variation, estimates of the new process mean ( ) could be expected to vary, introducing uncertainty into the determination of . Such uncertainty is a function of sample size. Suppose that wafers were used to estimate . Because of the random wafer-to-wafer variation, the resulting sample mean ( ) would not equal exactly. Processing and measuring groups of wafers repeatedly would produce values for which would vary about with a standard deviation of This result is true regardless of the nature of the parent distribution; it does not need to be normal.