Higher Engineering Mathematics

SIGNIFICANCE TESTING 593

J

and standard deviation are part of a normal distri-
bution. For a set of results to be probably significant
a confidence level of 95% is required for a particular
hypothesis being probably correct. This is equivalent
to the hypothesis being rejected when the level of sig-
nificance is greater than 0.05. For this to occur, the
z-value of the mean of the samples will lie between
−1.96 and+1.96 (since the area under the stan-
dardised normal distribution curve between these
z-values is 95%). The shaded area in Fig. 62.1 is
based on results which are probably significant, i.e.
having a level of significance of 0.05, and represents
the probability of rejecting a hypothesis when it is
correct. Thez-values of less than−1.96 and more
than 1.96 are calledcritical valuesand the shaded
areas in Fig. 62.1 are called thecritical regionsor
regions for which the hypothesis is rejected. Having
formulated hypotheses, the rules of decision and a
level of significance, the magnitude of the type I
error is given. Nothing can now be done about type
II errors and in most cases they are accepted in the
hope that they are not too large.

−1.96 1.96

95 % of total area Critical region(2.5% of

total area)

Critical region
(2.5% of
total area)

z

Figure 62.1

When critical regions occur on both sides of the
mean of a normal distribution, as shown in Fig. 62.1,
they are as a result oftwo-tailedortwo-sided tests.
In such tests, consideration has to be given to val-
ues on both sides of the mean. For example, if it is
required to show that the percentage of metal,p,ina
particular alloy isx%, then a two-tailed test is used,
since the null hypothesis is incorrect if the percent-
age of metal is either less thanxor more thanx. The
hypothesis is then of the form:

H 0 :p=x% H 1 :p=x%

However, for the machine producing bolts, the man-
ufacturer’s decision is not affected by the fact that
a sample contains say 1 or 2 defective bolts. He is
only concerned with the sample containing, say, 10
or more effective bolts. Thus a ‘tail’ on the left of the
mean is not required. In this case aone tailed test
or aone-sided testis really required. If the defect
rate is, say,dand the per unit values economically

acceptable to the manufacturer areu 1 andu 2 , where

u 1 is an acceptable defect rate andu 2 is the maxi-

mum acceptable defect rate, then the hypotheses in

this case are of the form:

H 0 :d=u 1 H 1 :d>u 2

and the critical region lies on the right-hand side of

the mean, as shown in Fig. 62.2(a). A one-tailed test

can have its critical region either on the right-hand

side or on the left-hand side of the mean. For exam-

ple, if lamps are being tested and the manufacturer

is only interested in those lamps whose life length

does not meet a certain minimum requirement, then

the hypotheses are of the form:

H 0 :l=hH 1 :l<h

wherelis the life length andhis the number of

hours to failure. In this case the critical region lies

on the left-hand side of the mean, as shown in

Fig. 62.2(b).

Critical region

(5% of

total area)

95 % of total area

1.645 z

Critical region

(5% of

total area)

95 % of total area

−1.645

(b)

(a)

Figure 62.2

Thez-values for various levels of confidence are

given are given in Table 61.1 on page 582. The cor-

responding levels of significance (a confidence level

of 95% is equivalent to a level of significance of 0.05

in a two-tailed test) and theirz-values for both one-

tailed and two-tailed tests are given in Table 62.1. It

can be seen that two values ofzare given for one-

tailed tests, the negative value for critical regions

lying to the left of the mean and a positive value for

critical regions lying to the right of the mean.