Page 370 - DMTH404_STATISTICS

Statistics

Notes
1.42 1.26 9 13
-
´
 The test statistic is z = = 0.16 ´ = 0.369.
1 1 22
+
9 13
Since this value is less than 1.96, there is no evidence against H at 5% level of significance. Thus,
0
the given samples provide no evidence of different correlations in two populations.
2
26.2 Uses of  test
In addition to the use of  in tests of hypothesis concerning the standard deviation, it is used as
2
a test of goodness of fit and as a test of independence of two attributes. These tests are explained
in the following sections.

2
26.2.1  - test as a Goodness of Fit
2
 - test can be used to test, how far the fitted or the expected frequencies are in agreement with
the observed frequencies. We know that for large values of n, the sampling distribution of X, the
number of successes, is normal with mean np and variance np(1 - ). Thus,
X n
-
z = ~ N (0,1 ).
n (1 - )
Further, square of z is a c - variate with one degree of freedom. We can write
2

(X n- ) 2 2 é (1 - )  ù
+
2
z = = (X n- ) ê ú ( 1 1  = - + )
n (1 - ) ë n (1 - ) û
2 é 1 1 ù (X n- ) 2 (X n- ) 2
(X n= - ) ê + ú = + .... (1)
ë n n (1 - ) û n n (1 - )
+
-
(X - n ) 2 (X - n n n ) 2 ( [ X - ) n + n (1 - )] 2
We can write = =
n (1 - ) n (1 - ) n (1 - )
( [ n X- ) n- (1 - )] 2 ( [ n X- ) E- (n X- )] 2
= =
n (1 - ) E (n X- )

( )]
(X - n ) 2 [X - E X 2
Similarly = .
( )
n E X
( )] [
[X - E X 2 (n X- ) E- (n X- )] 2
2
Thus, equation (1) can be written as z = +
( )
E X E (n X- )
Here X denotes the observed number of successes and (n - X) the observed number of failures.
Let O , E denote the observed and expected number of successes respectively and O , E denote
1 1 2 2
the observed and expected number of failures respectively.

(O - E ) 2 (O - E ) 2
2
2
 z = 1 1 + 2 2 is a  - variate with 1 d.f.
E E
1 2

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