Page 36 - Applied statistics and probability for engineers

14 Chapter 1/The Role of Statistics in Engineering

More details are useful to describe the role of probability models. Suppose that a produc-
tion lot contains 25 wafers. If all the wafers are defective or all are good, clearly any sample
will generate all defective or all good wafers, respectively. However, suppose that only 1 wafer
in the lot is defective. Then a sample might or might not detect (include) the wafer. A prob-
ability model, along with a method to select the sample, can be used to quantify the risks that
the defective wafer is or is not detected. Based on this analysis, the size of the sample might
be increased (or decreased). The risk here can be interpreted as follows. Suppose that a series
of lots, each with exactly one defective wafer, is sampled. The details of the method used to
select the sample are postponed until randomness is discussed in the next chapter. Neverthe-
less, assume that the same size sample (such as three wafers) is selected in the same manner
from each lot. The proportion of the lots in which the defective wafer are included in the sam-
ple or, more speciically, the limit of this proportion as the number of lots in the series tends to
ininity, is interpreted as the probability that the defective wafer is detected.
A probability model is used to calculate this proportion under reasonable assumptions
for the manner in which the sample is selected. This is fortunate because we do not want to
attempt to sample from an ininite series of lots. Problems of this type are worked in Chapters
2 and 3. More importantly, this probability provides valuable, quantitative information regard-
ing any decision about lot quality based on the sample.
Recall from Section 1-1 that a population might be conceptual, as in an analytic study
that applies statistical inference to future production based on the data from current pro-
duction. When populations are extended in this manner, the role of statistical inference
and the associated probability models become even more important.
In the previous example, each wafer in the sample was classiied only as defective or
not. Instead, a continuous measurement might be obtained from each wafer. In Section
1-2.5, concentration measurements were taken at periodic intervals from a production
process. Figure 1-8 shows that variability is present in the measurements, and there might
be concern that the process has moved from the target setting for concentration. Similar
to the defective wafer, one might want to quantify our ability to detect a process change
based on the sample data. Control limits were mentioned in Section 1-2.5 as decision rules
for whether or not to adjust a process. The probability that a particular process change
is detected can be calculated with a probability model for concentration measurements.
Models for continuous measurements are developed based on plausible assumptions for
the data and a result known as the central limit theorem, and the associated normal dis-
tribution is a particularly valuable probability model for statistical inference. Of course,
a check of assumptions is important. These types of probability models are discussed in
Chapter 4. The objective is still to quantify the risks inherent in the inference made from
the sample data.
Throughout Chapters 6 through 15, we base decisions on statistical inference from sample
data. We use continuous probability models, speciically the normal distribution, extensively
to quantify the risks in these decisions and to evaluate ways to collect the data and how large
a sample should be selected.

Important Terms and Concepts

Analytic study Fractional factorial Overcontrol Scientiic method
Cause and effect experiment Population Statistical inference
Designed experiment Hypothesis Probability model Statistical process control
Empirical model Hypothesis testing Random variable Statistical thinking
Engineering method Interaction Randomization Tampering
Enumerative study Mechanistic model Retrospective study Time series
Factorial experiment Observational study Sample Variability