Higher Engineering Mathematics

592 STATISTICS AND PROBABILITY

Using Table 58.1 of partial areas under the stan-

dardised normal curve given on page 561, az-value

of−0.32 corresponds to an area between the mean

and the ordinate atz=− 0 .32 to 0.1255. Thus, the

probability of there being 9 or less defective bolts

in the sample is given by the area to the left of

thez= 0 .32 ordinate, i.e. 0. 5000 − 0 .1255, that is,

0.3745. Thus, the probability of getting 9 or less

defective bolts in a sample of 200 bolts,even though

the defect rate has risen to 5%, is 37%. It follows

that as a result of the manufacturer’s decisions, for

37 samples in every 100, the machine will be left

running even though the defect rate has risen to 5%.

In general terms:

‘a hypothesis has been accepted when it should have

been rejected’.

When this occurs, it is called atype II error, and,
in this example, the type II error is 37%.
Tests of hypotheses and rules of decisions should
be designed to minimise the errors of decision. This
is achieved largely by trial and error for a particular
set of circumstances. Type I errors can be reduced by
increasing the number of defective items allowable
in a sample, but this is at the expense of allowing a
larger percentage of defective items to leave the fac-
tory, increasing the criticism from customers. Type II
errors can be reduced by increasing the percent-
age defect rate in the alternative hypothesis. If a
higher percentage defect rate is given in the alter-
native hypothesis, the type II errors are reduced very
effectively, as shown in the second of the two tables
below, relating the decision rule to the magnitude
of the type II errors. Some examples of the magni-
tude of type I errors are given below, for a sample
of 1000 components being produced by a machine
with a mean defect rate of 5%.

Decision rule Type I error

Stop production if the number (%)

of defective components is

equal to or greater than:

52 38.6

56 19.2

60 7.35

64 2.12

68 0.45

The magnitude of the type II errors for the output of
the same machine, again based on a random sample
of 1000 components and a mean defect rate of 5%,
is given below.

Decision rule Type II error

Stop production when (%)

the number of defective

components is 60, when the

defect rate is (%):

5.5 75.49

7 10.75

8.5 0.23

10 0.00

When testing a hypothesis, the largest value of

probability which is acceptable for a type I error is

called thelevel of significanceof the test. The level

of significance is indicated by the symbolα(alpha)

and the levels commonly adopted are 0.1, 0.05, 0.01,

0.005 and 0.002. A level of significance of, say, 0.05

means that 5 times in 100 the hypothesis has been

rejected when it should have been accepted.

In significance tests, the following terminology is

frequently adopted:

(i) if the level of significance is 0.01 or less, i.e.

the confidence level is 99% or more, the results

are considered to behighly significant, i.e. the

results are considered likely to be correct,

(ii) if the level of significance is 0.05 or between

0.05 and 0.01, i.e. the confidence level is 95% or

between 95% and 99%, the results are consid-

ered to beprobably significant, i.e. the results

are probably correct,

(iii) if the level of significance is greater than 0.05,

i.e. the confidence level is less than 95%, the

results are considered to benot significant, that

is, there are doubts about the correctness of the

results obtained.

This terminology indicates that the use of a level of

significance of 0.05 for ‘probably significant’ is, in

effect, a rule of thumb. Situations can arise when the

probability changes with the nature of the test being

done and the use being made of the results.

The example of a machine producing bolts, used

to illustrate type I and type II errors, is based on a

single random sample being drawn from the output

of the machine. In practice, sampling is a continu-

ous process and using the data obtained from several

samples, sampling distributions are formed. From

the concepts introduced in Chapter 61, the means

and standard deviations of samples are normally

distributed, thus for a particular sample its mean