Page 371 - DMTH404_STATISTICS

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Unit 26:  - Test Hypothesis

Also we note that O + O = E + E = n. Notes
1 2 1 2
The above result can be generalised for a manifold classification. If a population is divided into
k mutually exclusive classes with observed and expected frequencies as O , O , ...... O and E , E ,
1 2 k 1 2
k (O - E ) 2
2
...... E respectively, then  = å i i is a c - variate with (k - 1) d.f. Here again we have
2
k E
i= 1 i
k k
å O = E = N (total frequency).
i å
i
i= 1 i= 1
Example 40: 300 digits were chosen from a table of numbers and the following frequency
distribution was obtained :

Digit : 0 1 2 3 4 5 6 7 8 9
Frequency : 26 28 33 32 28 37 33 30 30 23

Test the hypothesis that the digits are uniformly distributed over the table.
Solution.
When H is true, the expected frequency of each digit would be 30.
0
1 2
2
  = å O - N
i
30
1 2 2 2 2 2 2 2 2 2 2
= (26 + 28 + 33 + 32 + 28 + 37 + 33 + 30 + 30 + 23 ) 300- = 4.8
30
2
The value of  from table for 5% level of significance and 9 d.f. is 16.92. Since the calculated value
is less than tabulated, there is no evidence against H . Thus, the distribution of numbers over the
0
table may be treated as uniform.

Example 41: A sample analysis of examination results of 200 M.B.A.'s was made. It was
found that 46 students had failed, 68 secured a third division, 62 secured a second division and
the rest were placed in the first division. Are these figures commensurate with the general
examination result which is in the ratio of 2 : 3 : 3 : 2 for the various categories, respectively?
(Given : Table value of chi-square for 3 d.f. at 5% level of significance is 7.81.)

Solution.
H : The students in various categories are distributed in the ratio 2 : 3 : 3 : 2.
0
The expected number of students, under the assumption that H is true, are :
0
2
expected number of failures = ´ 200 = 40 ,
(2 3 3 2+ + + )

3
expected number of third divisioners = ´ 200 = 60 ,
10
3
expected number of second divisioners = ´ 200 = 60 and
10

2
expected number of first divisioners = ´ 200 = 40 .
10

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