Higher Engineering Mathematics

598 STATISTICS AND PROBABILITY

The particular value ofxof a large sample drawn

for a significance test is therefore part of a normal

distribution and it is possible to determine by how

muchx is likely to differ fromμxin terms of

the normal standard variatez. The relationship is

z=

x−μx

σx

.

However, with reference to Chapter 61, page 578,

σx=

σ

√

N

√(

Np−N

Np− 1

)

for finite populations,

=

σ

√

N

for infinite populations, andμx=μ

whereNis the sample size,Npis the size of the

population,μis the mean of the population andσ

the standard deviation of the population.

Substituting forμx andσxin the equation forz

gives:

z=

x−μ

σ

√

N

for infinite populations, (1)

z=

x−μ

σ

√

N

√(

Np−N

Np− 1

) (2)

for populations of sizeNp

In Table 62.1 on page 594, the relationship bet-
weenz-values and levels of significance for both
one-tailed and two-tailed tests are given. It can be
seen from this table for a level of significance of,
say, 0.05 and a two-tailed test, thez-value is+1.96,
andz-values outside of this range are not signifi-
cant. Thus, for a given level of significance (i.e. a
known value ofz), the mean of the population,μ, can
be predicted by using equations (1) and (2) above,
based on the mean of a samplex. Alternatively, if the
mean of the population is known, the significance of
a particular value ofz, based on sample data, can be
established. If thez-value based on the mean of a ran-
dom sample for a two-tailed test is found to be, say,
2.01, then at a level of significance of 0.05, that is, the
results being probably significant, the mean of the
sampling distribution is said to differ significantly
from what would be expected as a result of the null
hypothesis (i.e. thatx=μ), due to the result of the
test being classed as ‘not significant’ (see page 592).
The hypothesis would then be rejected and an alter-
native hypothesis formed, i.e.H 1 :x=μ. The rules

of decision for such a test would be:

(i) reject the hypothesis at a 0.05 level of signifi-

cance, i.e. if thez-value of the sample mean is

outside of the range−1.96 to+1.96.

(ii) accept the hypothesis otherwise.

For small sample sizes (usually taken asN<30),

the sampling distribution is not normally distributed,

but approximates to Student’st-distributions (see

Section 61.5). In this case, t-values rather than

z-values are used and the equations analogous to

equations (1) and (2) are:

|t|=

x−μ

σ

√

N

for infinite populations (3)

|t|=

x−μ

σ

√

N

√(

Np−N

Np− 1

) (4)

for populations of sizeNp

where|t|means the modulus oft, i.e. the positive

value oft.

(b) When the standard deviation of the

population is not known

It is found, in practice, that if the standard devia-

tion of a sample is determined, its value is less than

the value of the standard deviation of the population

from which it is drawn. This is as expected, since

the range of a sample is likely to be less than the

range of the population. The difference between the

two standard deviations becomes more pronounced

when the sample size is small. Investigations have

shown that the variance,s^2 , of a sample ofNitems

is approximately related to the variance,σ^2 ,ofthe

population from which it is drawn by:

s^2 =

(

N− 1

N

)

σ^2

The factor

(

N− 1

N

)

is known asBessel’s cor-

rection. This relationship may be used to find the

relationship between the standard deviation of a

sample,s, and an estimate of the standard deviation

of a population,σˆ, and is:

σˆ^2 =s^2

(

N

N− 1

)

i.e.σˆ=s

√(

N

N− 1

)