Higher Engineering Mathematics

610 STATISTICS AND PROBABILITY

Number of Observed Expected

boys (B) frequency, frequency,

and girls (G) oe

5B, 0G 11 6

4B, 1G 35 31

3B, 2G 69 63

2B, 3G 55 63

1B, 4G 25 31

0B, 5G 5 6

o−e (o−e)^2

(o−e)^2

e

5 25 4.167

4 16 0.516

6 36 0.571

− 8 64 1.016

− 6 36 1.161

− 1 1 0.167

χ^2 =

∑

{

(o−e)^2

e

}

= 7. 598

To test the significance of theχ^2 -value

The number of degrees of freedom is given by
ν=N−1 whereN is the number of rows in the
table above, thusν= 6 − 1 =5. For a level of sig-
nificance of 0.05, the confidence level is 95%, i.e.
0.95 per unit. From Table 63.1 for theχ^20. 95 ,ν= 5
value, the percentile valueχ^2 pis 11.1. Since the calcu-
lated value ofχ^2 is less thanχ^2 pthe null hypothesis
that the observed frequencies are consistent with
male and female births being equally probable is
accepted.
For a confidence level of 95%, theχ^20. 05 ,ν= 5
value from Table 63.1 is 1.15 and because the cal-
culated value ofχ^2 (i.e. 7.598) is greater than this
value,the fit is not so good as to be unbelievable.

Problem 3. The deposition of grit particles

from the atmosphere is measured by counting

the number of particles on 200 prepared cards

in a specified time. The following distribution

was obtained.

Number of

particles 0123456

Number

of cards 41 69 44 27 12 6 1

Test the null hypothesis that the deposition of grit

particles is according to a Poisson distribution at

a level of significance of 0.01 and determine if

the data is ‘too good to be true’ at a confidence

level of 99%.

To determine the expected frequency

The expectation or average occurrence is given by:

λ=

total number of particles deposited

total number of cards

=

69 + 88 + 81 + 48 + 30 + 6

200

= 1. 61

The expected frequencies are calculated using

a Poisson distribution, where the probabili-

ties of there being 0, 1, 2,..., 6 particles

deposited are given by the successive terms of

e−λ

(

1 +λ+

λ^2

2!

+

λ^3

3!

+ ···

)

taken from left to

right,

i.e. e−λ,λe−λ,

λ^2 e−λ

2!

,

λ^3 e−λ

3!

···

Calculating these terms forλ= 1 .61 gives:

Number of

particles Expected

deposited Probability frequency

0 0.1999 40

1 0.3218 64

2 0.2591 52

3 0.1390 28

4 0.0560 11

5 0.0180 4

6 0.0048 1

To determine theχ^2 -valve

Theχ^2 -value is calculated using a tabular method as

shown below.

Number of Observed Expected

grit particles frequency,o frequency,e

04140

16964

24452

32728

41211

564

611