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4-6 NORMAL DISTRIBUTION 109

(b) Determine the cumulative distribution function of the (a) What percentage of tests require more than 70 seconds to
weight of packages. complete.
(c) Determine P1X 50.12. (b) What percentage of tests require less than one minute to
4-34. The thickness of a flange on an aircraft component is complete.
uniformly distributed between 0.95 and 1.05 millimeters. (c) Determine the mean and variance of the time to complete
(a) Determine the cumulative distribution function of flange a test on a sample.
thickness. 4-37. The thickness of photoresist applied to wafers in
(b) Determine the proportion of flanges that exceeds 1.02 semiconductor manufacturing at a particular location on the
millimeters. wafer is uniformly distributed between 0.2050 and 0.2150
(c) What thickness is exceeded by 90% of the flanges? micrometers.
(d) Determine the mean and variance of flange thickness. (a) Determine the cumulative distribution function of pho-
4-35. Suppose the time it takes a data collection operator to toresist thickness.
fill out an electronic form for a database is uniformly between (b) Determine the proportion of wafers that exceeds 0.2125
1.5 and 2.2 minutes. micrometers in photoresist thickness.
(a) What is the mean and variance of the time it takes an op- (c) What thickness is exceeded by 10% of the wafers?
erator to fill out the form? (d) Determine the mean and variance of photoresist thickness.
(b) What is the probability that it will take less than two min- 4-38. The probability density function of the time required
utes to fill out the form? to complete an assembly operation is f 1x2 0.1 for
(c) Determine the cumulative distribution function of the time 30 x 40 seconds.
it takes to fill out the form. (a) Determine the proportion of assemblies that requires more
4-36. The probability density function of the time it takes a than 35 seconds to complete.
hematology cell counter to complete a test on a blood sample (b) What time is exceeded by 90% of the assemblies?
is f 1x2 0.2 for 50 x 75 seconds. (c) Determine the mean and variance of time of assembly.

4-6 NORMAL DISTRIBUTION

Undoubtedly, the most widely used model for the distribution of a random variable is a normal
distribution. Whenever a random experiment is replicated, the random variable that equals the
average (or total) result over the replicates tends to have a normal distribution as the number of
replicates becomes large. De Moivre presented this fundamental result, known as the central
limit theorem, in 1733. Unfortunately, his work was lost for some time, and Gauss independ-
ently developed a normal distribution nearly 100 years later. Although De Moivre was later
credited with the derivation, a normal distribution is also referred to as a Gaussian distribution.
When do we average (or total) results? Almost always. For example, an automotive engi-
neer may plan a study to average pull-off force measurements from several connectors. If we
assume that each measurement results from a replicate of a random experiment, the normal
distribution can be used to make approximate conclusions about this average. These conclu-
sions are the primary topics in the subsequent chapters of this book.
Furthermore, sometimes the central limit theorem is less obvious. For example, assume that
the deviation (or error) in the length of a machined part is the sum of a large number of in-
finitesimal effects, such as temperature and humidity drifts, vibrations, cutting angle variations,
cutting tool wear, bearing wear, rotational speed variations, mounting and fixturing variations,
variations in numerous raw material characteristics, and variation in levels of contamination. If
the component errors are independent and equally likely to be positive or negative, the total error
can be shown to have an approximate normal distribution. Furthermore, the normal distribution
arises in the study of numerous basic physical phenomena. For example, the physicist Maxwell
developed a normal distribution from simple assumptions regarding the velocities of molecules.
The theoretical basis of a normal distribution is mentioned to justify the somewhat com-
plex form of the probability density function. Our objective now is to calculate probabilities
for a normal random variable. The central limit theorem will be stated more carefully later.