CHI-SQUARE AND DISTRIBUTION-FREE TESTS 613
J
- The data given below refers to the num-
ber of people injured in a city by accidents
for weekly periods throughout a year. It is
believed that the data fits a Poisson distri-
bution. Test the goodness of fit at a level of
significance of 0.05.
Number of Number of
people injured weeks
in the week
05
112
213
39
47
54
62
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λ= 2 .404; expected
frequencies: 11, 27, 33, 26, 16, 8, 3
χ^2 -value= 42 .24;
χ^20. 95 ,ν 6 = 12 .6, hence the data
does not fit a Poisson distribution
at a level of significance of 0.05
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- The resistances of a sample of carbon resis-
tors are as shown below.
Resistance Frequency
(M)
1.28 7
1.29 19
1.30 41
1.31 50
1.32 73
1.33 52
1.34 28
1.35 17
1.36 9
Test the null hypothesis that this data corre-
sponds to a normal distribution at a level of
significance of 0.05.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x ̄= 1 .32,s= 0 .0180; expected
frequencies, 6, 17, 36, 55, 65,
55, 36, 17, 6;χ^2 -value= 5 .98;
χ^20. 95 ,ν 6 = 12 .6, hence the
null hypothesis is accepted, i.e.
the data does correspond to a
normal distribution
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- The quality assurance department of a firm
selects 250 capacitors at random from a large
quantity of them and carries out various tests
on them. The results obtained are as follows:
Number of Number of
tests failed capacitors
0 113
177
239
316
44
51
6 and over 0
Test the goodness of fit of this distribution to a
Poisson distribution at a level of significance
of 0.05.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λ= 0 .896; expected
frequencies are 102, 91, 41,
12, 3, 0, 0;χ^2 -value= 5. 10.
χ^20. 95 ,ν 6 = 12 .6, hence this
data fits a Poisson distribution
at a level of significance of 0.05
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Test the hypothesis that the maximum load
before breaking supported by certain cables
produced by a company follows a normal dis-
tribution at a level of significance of 0.05,
based on the experimental data given below.
Also test to see if the data is ‘too good’ at a
level of significance of 0.05.
Maximum Number of
load (MN) cables
8.5 2
9.0 5
9.5 12
10.0 17
10.5 14
11.0 6
11.5 3
12.0 1
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x ̄= 10 .09 MN;σ= 0 .733 MN;
expected frequencies, 2, 5, 12,
16, 14, 8, 3, 1;χ^2 -value= 0 .563;
χ^20. 95 ,ν 5 = 11. 1 .Hence
hypothesis accepted.χ^20. 05 ,
ν 5 = 1 .15, hence the results are
‘too good to be true’
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
63.3 Introduction to distribution-free
tests
Sometimes, sampling distributions arise from pop-
ulations with unknown parameters. Tests that deal
