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450 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
Since 1.83 is less than the critical value of z = 1.96, we are unable to reject H 0 . We
conclude that the population correlation coefficient may be .80.
For sample sizes less than 25, Fisher’s Z transformation should be used with caution,
if at all. An alternative procedure from Hotelling (6) may be used for sample sizes
equal to or greater than 10. In this procedure the following transformation of r is employed:
The standard deviation of z* is
The test statistic is
where
Z* =
(9.7.7)
(9.7.8)
(9.7.9)
Critical values for comparison purposes are obtained from the standard normal
distribution.
In our present example, to test H 0 : r = .80 against H A : r Z .80 using the
Hotelling transformation and a = .05, we have
z* = 1.24726 -
z* = 1.09861 -
z* = z r - 3z r + r
4n
s z* =
z* - z*
1> 1n - 1
1
1n - 1
= 1z* - z*2 1n - 1
z* 1pronounced zeta2 = z r - 13z r + r2
4n
311.247262 + .848
411552
311.098612 + .8
411552
= 1.2339
= 1.0920
Z* = 11.2339 - 1.09202 1155 - 1 = 1.7609
Since 1.7609 is less than 1.96, the null hypothesis is not rejected, and the same conclusion
is reached as when the Fisher transformation is used.
Alternatives In some situations the data available for analysis do not meet the assumptions
necessary for the valid use of the procedures discussed here for testing hypotheses about
a population correlation coefficient. In such cases it may be more appropriate to use the Spearman
rank correlation technique discussed in Chapter 13.
Confidence Interval for R Fisher’s transformation may be used to construct
10011 - a2 percent confidence intervals for r. The general formula for a confidence
interval
estimator ; (reliability factor)(standard error)