Page 145 - Statistics for Environmental Engineers

L1592_Frame_C16 Page 142 Tuesday, December 18, 2001 1:51 PM
y
We do not expect to observe that = η 0 , even if η = η 0 . However, if is near η 0 , it can reasonably
y
be concluded that η = η 0 and that the measured value agrees with the specified value. Therefore, some
statement is needed as to how close we can reasonably expect the estimate to be. If the process is on-
– will fall within bounds that are a multiple of the standard
standard or on-specification, the distance y η 0
deviation of the measurements.
We make use of the fact that for n < 30,
y η–
t = ------------
s/ n

is a random variable which has a t distribution with ν = n − 1 degrees of freedom. s is the sample standard
deviation. Consequently, we can assert, with probability 1 − α, that the inequality:
y η
–
– t ν,α /2 ≤ ------------ ≤ t ν,α /2
s/ n
will be satisfied. This means that the maximum value of the error y – η is:

s
y – η = t ν,α /2 -------
n

with probability 1 − α. In other words, we can assert with probability 1 − α that the error in using y
s
------- .
to estimate η will be at most t ν,α /2 n
From here, the comparison of the estimated mean with the standard value can be done as a hypothesis
test or by computing a confidence interval. The two approaches are equivalent and will lead to the same
conclusion. The confidence interval approach is more direct and often appeals to engineers.

Testing the Null Hypothesis

The comparison between and η 0 can be stated as a null hypothesis:
y
H 0 : η – η 0 = 0
which is read “the expected difference between η and η 0 is zero.” The “null” is the zero. The extent to
which differs from η will be due to only random measurement error and not to bias. The extent
y
to which differs from η 0 will be due to both random error and bias. We hypothesize the bias (η − η 0 ) toy
be zero, and test for evidence to the contrary.
The sample average is:

y = ∑y i
--------
n
The sample variance is:

(
s = ∑ y i – y)
2
----------------------
–
n 1
and the standard error of the mean is:
s
s y = -------
n

© 2002 By CRC Press LLC