Page 34 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 23

when the die comes to rest. The probabilistic nature of some events is apparent when ques-
tions of the following type are asked. Does medicine A cure a disease better than medicine
B? What is the ultimate strength of 1020 steel? What total mileage will brand X tire yield?
Which heat treatment process is better for a given part?
Answering such questions involves designing experiments, performing measurements,
analyzing the data, and interpreting the results. In this endeavor two common phenomena
are observed: (1) repeated measurements of the same attribute differ due to measurement
error and resolving capability of the measurement system and (2) corresponding attributes
of identical entities differ due to material differences, manufacturing tolerances, tool wear,
and so on.
Conclusions based on experiments are statistical inferences and can only be made with
some element of doubt. Experiments are performed to make statistical inferences with min-
imum doubt. Therefore, experiments are designed specifying the data required, amount of
data needed, and the confidence limits desired in drawing conclusions. In this process an
instrumentation system is specified, a data-taking procedure is outlined, and a statistical
method is used to make conclusions at preselected confidence levels.
In statistical analysis of experimental data, the descriptive and inference tasks are con-
sidered. The descriptive task is to present a comprehensible set of observations. The inference
task determines the truth of the whole by examination of a sample. The inference task
requires sampling, comparison, and a variety of statistical tests to obtain unbiased estimates
and confidence limits to make decisions.

Statistical Testing
A statistical hypothesis is an assertion relative to the distribution of a random variable. The
test of a statistical hypothesis is a procedure to accept or reject the hypothesis. A hypothesis
is stated such that the experiment attempts to nullify the hypothesis. Therefore, the hypothesis
under test is called the null hypothesis and symbolized by H . All alternatives to the null
0
hypothesis are termed the alternate hypothesis and are symbolized by H . 15
1
If the results of the experiment cannot reject H , the experiment cannot detect the dif-
0
ferences in measurements at the chosen probability level.
Statistical testing determines if a process or item is better than another with some stated
degree of confidence. The concept can be used with a certain statistical distribution to de-
termine the confidence limits.
The following procedure is used in statistical testing:
1. Define H and H .
1
0
2. Choose the confidence level of the test.
3. Form an appropriate probabilistic statement.
4. Using the appropriate statistical distribution, perform the indicated calculation.
5. Make a decision concerning the hypothesis and/or determine confidence limits.

Two types of error are possible in statistical testing. A Type I error is that of rejecting
a true null hypothesis (rejecting truth). A Type II error is that of accepting a false null
hypothesis (embracing fiction). The confidence levels and sample size are chosen to minimize
the probability of making a Type I or Type II error. Figure 5 illustrates the Type I ( ) and
Type II (
) errors, where

H sample with n 1 comes from N(10,16)
0
H sample with n 1 comes from N(17,16)
1