Statistics and probability
62
Significance testing
62.1 Hypotheses
Industrial applications of statistics is often con-
cerned with making decisions about populations
and population parameters. For example, decisions
about which is the better of two processes or deci-
sions about whether to discontinue production on a
particular machine because it is producing an eco-
nomically unacceptable number of defective com-
ponents are often based on deciding the mean
or standard deviation of a population, calculated
using sample data drawn from the population.
In reaching these decisions, certain assumptions
are made, which may or may not be true. The
assumptions made are calledstatistical hypothe-
sesor justhypothesesand are usually concerned
with statements about probability distributions of
populations.
For example, in order to decide whether a dice
is fair, that is, unbiased, a hypothesis can be made
that a particular number, say 5, should occur with a
probability of one in six, since there are six numbers
on a dice. Such a hypothesis is called anull hypoth-
esisand is an initial statement. The symbolH 0 is
used to indicate a null hypothesis. Thus, ifpis the
probability of throwing a 5, thenH 0 :p=^16 means,
‘the null hypothesis that the probability of throw-
inga5is^16 ’. Any hypothesis which differs from a
given hypothesis is called analternative hypothe-
sis, and is indicated by the symbolH 1. Thus, if after
many trials, it is found that the dice is biased and
that a 5 only occurs, on average, one in every seven
throws, then several alternative hypotheses may be
formulated. For example:H 1 :p=^17 orH 1 :p<^16 or
H 1 :p>^18 orH 1 :p=^16 are all possible alternative
hypotheses to the null hypothesis thatp=^16.
Hypotheses may also be used when comparisons
are being made. If we wish to compare, say, the
strength of two metals, a null hypothesis may be for-
mulated that there isno differencebetween the
strengths of the two metals. If the forces that the two
metals can withstand areF 1 andF 2 , then the null
hypothesis isH 0 :F 1 =F 2. If it is found that the
null hypothesis has to be rejected, that is, that thestrengths of the two metals are not the same, then the
alternative hypotheses could be of several forms. For
example,H 1 :F 1 >F 2 orH 1 :F 2 >F 1 orH 1 :F 1 =F 2.
These are all alternative hypotheses to the original
null hypothesis.62.2 Type I and Type II errors
To illustrate what is meant by type I and type II errors,
let us consider an automatic machine producing, say,
small bolts. These are stamped out of a length of
metal and various faults may occur. For example, the
heads or the threads may be incorrectly formed, the
length might be incorrect, and so on. Assume that,
say, 3 bolts out of every 100 produced are defec-
tive in some way. If a sample of 200 bolts is drawn
at random, then the manufacturer might be satisfied
that his defect rate is still 3% provided there are 6
defective bolts in the sample. Also, the manufacturer
might be satisfied that his defect rate is 3% or less
provided that there are 6 or less bolts defective in
the sample. He might then formulate the following
hypotheses:H 0 :p= 0 .03 (the null hypothesis that
the defect rate is 3%)The null hypothesis indicates that a 3% defect rate is
acceptable to the manufacturer. Suppose that he also
makes a decision that should the defect rate rise to
5% or more, he will take some action. Then the
alternative hypothesis is:H 1 :p≥ 0 .05 (the alternative hypothesis that
the defect rate is equal to or
greater than 5%)The manufacturer’s decisions, which are related to
these hypotheses, might well be:(i) a null hypothesis that a 3% defect rate is accept-
able, on the assumption that the associated num-
ber of defective bolts is insufficient to endanger
his firm’s good name;