Research Methodology - Dr. Krishan K. Pandey

Testing of Hypotheses I 225
the values happen to be positive; one must simply know the degrees of freedom for using such a
distribution. *
TESTING THE EQUALITY OF VARIANCES
OF TWO NORMAL POPULATIONS
When we want to test the equality of variances of two normal populations, we make use of F-test
2 2 2
based on F-distribution. In such a situation, the null hypothesis happens to be H0: σp = σ
1 p , σ
2 p1
2
and σ p2 representing the variances of two normal populations. This hypothesis is tested on the basis
2 2 2
of sample data and the test statistic F is found, using σ and σ the sample estimates for σ and
s1 s2
p1
2
σ p2 respectively, as stated below:
d i
F = σ
σ
X i X
X i X
2 1 1
where σs = σ
1 s2
n 1 n 1
∑ −
=
−
∑ −
and
−
b 1 g
2
2
s1
2
s
2 2 2
b 2 g
2
d i
2 2
While calculating F, σs1 is treated > σ which means that the numerator is always the greater
s2
variance. Tables for F-distribution ** have been prepared by statisticians for different values of F at
different levels of significance for different degrees of freedom for the greater and the smaller
variances. By comparing the observed value of F with the corresponding table value, we can infer
whether the difference between the variances of samples could have arisen due to sampling
fluctuations. If the calculated value of F is greater than table value of F at a certain level of significance
for (n – 1) and (n – 2) degrees of freedom, we regard the F-ratio as significant. Degrees of
1 2
freedom for greater variance is represented as v and for smaller variance as v . On the other hand,
1 2
if the calculated value of F is smaller than its table value, we conclude that F-ratio is not significant.
If F-ratio is considered non-significant, we accept the null hypothesis, but if F-ratio is considered
significant, we then reject H (i.e., we accept H ).
0 a
When we use the F-test, we presume that
(i) the populations are normal;
(ii) samples have been drawn randomly;
(iii) observations are independent; and
(iv) there is no measurement error.
The object of F-test is to test the hypothesis whether the two samples are from the same normal
population with equal variance or from two normal populations with equal variances. F-test was
initially used to verify the hypothesis of equality between two variances, but is now mostly used in the
* See Chapter 10 entitled Chi-square test for details.
** F-distribution tables [Table 4(a) and Table 4(b)] have been given in appendix at the end of the book.
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