Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 719

ESSENTIAL FORMULAE 719

Chi-square distribution

Percentile values (χ^2 p) for the Chi-square distribu-

tion withνdegrees of freedom—see Table 63.1 on

page 609.

χ^2 =

∑

{

(o−e)^2

e

}

whereoandeare the observed

and expected frequencies.

Symbols:

Population

number of membersNp, meanμ, standard devia-

tionσ.

Sample

number of membersN, meanx, standard deviations.

Sampling distributions

mean of sampling distribution of meansμx

standard error of meansσx

standard error of the standard deviationsσs.

Standard error of the means

Standard error of the means of a sample distribu-

tion, i.e. the standard deviation of the means of

samples, is:

σx=

σ

√

N

√(

Np−N

Np− 1

)

for a finite population and/or for sampling without

replacement, and

σx=

σ

√

N

for an infinite population and/or for sampling with

replacement.

The relationship between sample mean and

population mean

μx=μfor all possible samples of sizeNare drawn

from a population of sizeNp.

Estimating the mean of a population (σknown)

The confidence coefficient for a large sample size,

(N≥30) iszcwhere:

Confidence Confidence

level % coefficientzc

99 2.58

98 2.33

96 2.05

95 1.96

90 1.645

80 1.28

50 0.6745

The confidence limits of a population mean based

on sample data are given by:

x±

zcσ

√

N

√(

Np−N

Np− 1

)

for a finite population of sizeNp, and by

x±

zcσ

√

N

for an infinite population

Estimating the mean of a population (σunknown)

The confidence limits of a population mean based

on sample data are given by:μx±zcσx.

Estimating the standard deviation of a population

The confidence limits of the standard deviation of a

population based on sample data are given by:

s±zcσs.

Estimating the mean of a population based on a

small sample size

The confidence coefficient for a small sample size

(N<30) is tc which can be determined using

Table 61.1, page 582. The confidence limits of a

population mean based on sample data is given by:

x±

tcs

√

(N−1)