Higher Engineering Mathematics

608 STATISTICS AND PROBABILITY

Face Observed Expected

frequency frequency

149 40

235 40

332 40

446 40

549 40

629 40

Determine theχ^2 -value for this distribution.

[10.2]

2. The numbers of telephone calls received by
   the switchboard of a company in 200 five-
   minute intervals are shown in the distribution
   below.

Number of Observed Expected

calls frequency frequency

01116

14442

25352

34642

42426

51214

676

732

Calculate theχ^2 -value for this data.

[3.16]

63.2 Fitting data to theoretical

distributions

For theoretical distributions such as the binomial,
Poisson and normal distributions, expected frequen-
cies can be calculated. For example, from the theory
of the binomial distribution, the probability of hav-
ing 0, 1, 2,...,ndefective items in a sample ofn
items can be determined from the successive terms
of (q+p)n, wherepis the defect rate andq= 1 −p.
These probabilities can be used to determine the
expected frequencies of having 0, 1, 2,...,ndefec-
tive items. As a result of counting the number of
defective items when sampling, the observed fre-
quencies are obtained. The expected and observed
frequencies can be compared by means of a Chi-
square test and predictions can be made as to whether
the differences are due to random errors, due to
some fault in the method of sampling, or due to the
assumptions made.
As for normal andtdistributions, a table is avail-
able for relating various calculated values ofχ^2 to

those likely because of random variations, at vari-

ous levels of confidence. Such a table is shown in

Table 63.1. In Table 63.1, the column on the left

denotes the number of degrees of freedom,ν, and

when theχ^2 -values refer to fitting data to theoreti-

cal distributions, the number of degrees of freedom

is usually (N−1), whereNis the number of rows

in the table from which χ^2 is calculated. How-

ever, when the population parameters such as the

mean and standard deviation are based on sample

data, the number of degrees of freedom is given

byν=N− 1 −M, whereMis the number of esti-

mated population parameters. An application of

this is shown in Problem 4.

The columns of the table headedχ^20. 995 ,χ^20. 99 ,

...give the percentile ofχ^2 -values corresponding

to levels of confidence of 99.5%, 99%,...(i.e. lev-

els of significance of 0.005, 0.01,...). On the far

right of the table, the columns headed...,χ^20. 01 ,

χ^20. 005 also correspond to levels of confidence of...

99%, 99.5%, and are used to predict the ‘too good

to be true’ type results, where the fit obtained is so

good that the method of sampling must be suspect.

The method in whichχ^2 -values are used to test the

goodness of fit of data to probability distributions is

shown in the following problems.

Problem 2. As a result of a survey carried out

of 200 families, each with five children, the dis-

tribution shown below was produced. Test the

null hypothesis that the observed frequencies

are consistent with male and female births being

equally probable, assuming a binomial distribu-

tion, a level of significance of 0.05 and a ‘too

good to be true’ fit at a confidence level of 95%.

Number of boys (B) Number of

and girls (G) families

5B, 0G 11

4B, 1G 35

3B, 2G 69

2B, 3G 55

1B, 4G 25

OB, 5G 5

To determine the expected frequencies

Using the usual binomial distribution symbols, let p

be the probability of a male birth andq= 1 −pbe

the probability of a female birth. The probabilities

of having 5 boys, 4 boys,..., 0 boys are given by the

successive terms of the expansion of (q+p)n. Since