Higher Engineering Mathematics

582 STATISTICS AND PROBABILITY

in the intervals:

mean±1 standard deviation,

mean±2 standard deviations,

or mean±3 standard deviations,

by using tables of the partial areas under the stan-

dardized normal curve given in Table 58.1 on

page 561. From this table, the area corresponding

to az-value of+1 standard deviation is 0.3413, thus

the area corresponding to±1 standard deviation

is 2×0.3413, that is, 0.6826. Thus the percentage

probability of a sampling statistic lying between the

mean±1 standard deviation is 68.26%. Similarly,

the probability of a sampling statistic lying between

the mean±2 standard deviations is 95.44% and

of lying between the mean±3 standard deviations

is 99.74%.

The values 68.26%, 95.44% and 99.74% are

called theconfidence levelsfor estimating a sam-

pling statistic. A confidence level of 68.26% is

associated with two distinct values, these being,

S−(1 standard deviation), i.e.S−σsand

S+(1 standard deviation), i.e.S+σs. These two

values are called theconfidence limitsof the esti-

mate and the distance between the confidence limits

is called theconfidence interval. A confidence inter-

val indicates the expectation or confidence of finding

an estimate of the population statistic in that interval,

based on a sampling statistic. The list in Table 61.1

is based on values given in Table 58.1, and gives

some of the confidence levels used in practice and

their associatedz-values; (some of the values given

are based on interpolation). When the table is used

in this context,z-values are usually indicated by ‘zc’

and are called theconfidence coefficients.

Table 61.1

Confidence level, % Confidence coefficient,zc

99 2.58

98 2.33

96 2.05

95 1.96

90 1.645

80 1.28

50 0.6745

Any other values of confidence levels and their asso-
ciated confidence coefficients can be obtained using
Table 58.1.

Problem 4. Determine the confidence coef-

ficient corresponding to a confidence level

of 98.5%.

98.5% is equivalent to a per unit value of 0.9850.

This indicates that the area under the standardized

normal curve between−zcand+zc, i.e. correspond-

ing to 2zc, is 0.9850 of the total area. Hence the

area between the mean value andzcis

0. 9850

2

i.e.

0.4925 of the total area. Thez-value correspond-

ing to a partial area of 0.4925 is 2.43 standard

deviations from Table 58.1. Thus,the confidence

coefficient corresponding to a confidence limit of

98.5% is 2.43.

(a) Estimating the mean of a population when the

standard deviation of the population is known

When a sample is drawn from a large population

whose standard deviation is known, the mean value

of the sample, x, can be determined. This mean

value can be used to make an estimate of the mean

value of the population,μ. When this is done, the

estimated mean value of the population is given as

lying between two values, that is, lying in the con-

fidence interval between the confidence limits. If a

high level of confidence is required in the estimated

value ofμ, then the range of the confidence interval

will be large. For example, if the required confidence

level is 96%, then from Table 61.1 the confidence

interval is from−zcto+zc, that is, 2× 2. 05 = 4. 10

standard deviations wide. Conversely, a low level

of confidence has a narrow confidence interval and a

confidence level of, say, 50%, has a confidence inter-

valof2×0.6745, that is 1.3490 standard deviations.

The 68.26% confidence level for an estimate of the

population mean is given by estimating that the pop-

ulation mean,μ, is equal to the same mean,x, and

then stating the confidence interval of the estimate.

Since the 68.26% confidence level is associated with

‘±1 standard deviation of the means of the sampling

distribution’, then the 68.26% confidence level for

the estimate of the population mean is given by:

x± 1 σx

In general, any particular confidence level can be

obtained in the estimate, by usingx±zcσx, where

zcis the confidence coefficient corresponding to the

particular confidence level required. Thus for a 96%

confidence level, the confidence limits of the pop-

ulation mean are given byx± 2. 05 σx. Since only

one sample has been drawn, the standard error of the

means,σx, is not known. However, it is shown in

Section 61.3 that

σx=

σ

√

N

√(

Np−N

Np− 1

)