Page 368 - DMTH404_STATISTICS

Statistics

Notes For small samples, the test statistic can be obtained from the sampling distribution of b. We note
that if r = 0, then b would also be zero.

b S 1 S S n - 2 n - 2
Therefore, = r × Y × = r × Y × X r = will follow t - distribution with
E
E
S . . ( ) b S X S . . ( ) b S X S Y 1 r 2 1 r 2
-
-
n - 2 1 r 2
-
( )
(n - 2) d.f. Hence, r can be taken as the test statistic. We note that . .S E r = .
-
1 r 2 n - 2
Therefore, 100(1 - a)% confidence limits of r can be written as r ± t S.E.(r).
a/2
Example 1: A random sample of 11 pairs of observations from a bivariate normal
population gave r = 0.29. Test the significance of correlation in population.
Solution.
We have to test H :  = 0 against H :   0.
0 a
9
t = 0.29 = 0.91.
cal 2
-
1 0.29
The value of t from tables at 5% level of significance and 9 d.f. is 2.26. Thus, there is no evidence
against H .
0
26.1.2 Test of Hypothesis concerning Correlation Coefficient using
Fisher's Z test

This test is applicable whether n is small or large. If r is correlation in sample, then its Fisher's
1 1 r
+
Z transformation is given by Z = log e .
2 1 r
-
Further, if r is correlation in population, its Fisher's Z transformation, denoted by x, is given by
1 1 
+
x = log e
2 1 
-
Fisher has shown that the sampling distribution of Z is approximately normal with mean x and
1
standard error . Thus, (Z x- ) n - 3 ~ N (0,1 ).
n - 3
Note: Since the values of Z and x are defined using e as the base of the logarithms, it is necessary
to convert them into logarithms with base 10 for calculation purposes. Accordingly, we write

+
1 1 r 1 1 r 1 1 r 1
+
+
Z = log = log ´ log 10 = log ´
-
2 e 1 r 2 10 1 r e 2 10 1 r log e
-
-
10
1 1 r 1 r
+
+
´
= ´ 2.3026 log = 1.1513log
-
-
2 10 1 r 10 1 r
1 
+
Similarly, we have x = 1.1513log
10
-
1 
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