The document discusses Z tests and the limitations of tests of significance. It describes how to perform a Z test to test the significance of a correlation coefficient in a small sample by transforming the coefficient into a Z value. It also explains how to use a Z test to test the significance of the difference between two sample correlation coefficients. Finally, it outlines some limitations of relying solely on significance testing, including that tests with small samples can lack power and tests with large samples can find even small effects to be significant.

1. Z TEST AND
LIMITATIONS OF TEST OF
SIGNIFICANCE

2. Z Test
Fisher has also given a method of testing the significance of the
correlation in small samples. In this method the co-efficient of
correlation in the sample of r is transformed into Z. The value of Z is
calculated by the formula
Z= ½ loge* 1+r
1-r
The standard error of Z is
SEZ = 1
n-3

3. Similarly the co-efficient of correlation in the universe or P can be
transformed into . The value of  is calculated in the same value of Z
here the co-efficient of correlation of the universe is taken into
account.
To test the significance of a co-efficient of correlation the difference
between the Z and  is calculated and the difference of relationship and
then the standard error of Z is then found out. If the difference is more
than thrice or twice the standard error is supposed to be significant.

4. Significance of the difference between two
sample co- efficients of correlation
Z test can be used to test of significance of the difference of two
sample co- efficients of correlation. The co- efficients of correlation are
first transformed into Z by the formula and then the standard error of
the difference of the two Z’s is calculated by the formula
Standard Error= Z1- Z2 = 1 + 1
n1-3 n2-3
If actual difference between the two Z’s is more than thrice or twice
the standard error of the difference it is said to be significant otherwise
not.

5. Limitations Of Test 0f Significance
There are issues with relying on significance testing for evaluating
• Significance test should be used when generalising from the sample to
population. Significance test should be used only in attempts to generalise the
sample’s results to a population.
• Significance tests conducted with low power can be misleading
Program evaluation studies with less than approximately 50 participants tend to
lack sufficient statistical power.
• Significance tests conducted with high power can be misleading
For studies involving large sample sizes (e.g., > ~ 400), a different problem occurs
with significance testing because even small effects are likely to become
significant, although these effects may be trivial. In these situations, more
attention should be paid to effect sizes than to statistical significance testing.

6. • When examining effects using small sample size, significant testing
can be misleading because its subject to type II errors. Contrary to
population opinion, statistical significance is not a direct indicator of
size of effect, but rather it is a function of sample size, effect size and
p level.
• When examining effects using large samples, significant test can be
misleading because even small or trival effects are likely to produce
statistically significant results. This can be dealt with by reporting and
significance test results.