The ppt gives the detailed F test aloong with solved example.

1. F – TEST & ANALYSIS OF
VARIANCE (ANOVA)

2. F - Test
• F -test is a test of hypothesis concerning two variances derived from two
samples.
• F-statistic is the ratio of two independent unbiased estimators of population
variances and expressed as:
•
• F= 1
2
/2
2
• n1 – 1 the degrees of freedom for numerator and n2- 1 the degrees of freedom for
denominator.
• F –table gives variance ratio values at different levels of significance at d= (n1-1) given
horizontally and degrees of freedom (d) = (n2-1) given vertically.
• Generally 1
2
is greater than 2
2
but if 2
2
is greater than 1
2
, in such cases the two
variances should be interchanged so that the value of ‘F’ is always greater than 1.

3. • If the F – ratio value is smaller than the table value, the null
hypothesis (H0) is accepted. It indicates that the samples are drawn
from the same population.
• If the calculated F value is greater than the table value, null
hypothesis (H0) is rejected and conclude that the standard deviations
in the two populations are not equal.

4. • Working Procedure
• Set up the null hypothesis (H0) 1
2
= 2
2
and alternative hypothesis (H1) =
1
2
2
2
• Calculate the variances of two samples and then calculate the F statistic i.e.,
• F= 1
2
/2
2
if 1
2
2
2
• Or F= 2
2
/1
2
if 2
2
1
2
• Take level of significance at 0.05
• Compare the compound F- value with the table value and degrees of
freedom (n1 – 1) horizontally and degrees of freedom (n2 -1) vertically.

5. • Assumptions of F test:
• Normality: The values in each group should be normally distributed.
• Independence of Error: Variation of each value around its own group
mean i.e., error should be independent of each value.
• Homogenity: The variances within each group should be equal for all
groups i.e. 1
2
=2
2
= 3
2
=…….n
2

6. • Uses
• F test to check - equality of population variances.
• To test the two independent samples (x and y) have been drawn from
the normal populations with the same variances (2
).
• Whether the two independent estimates of the populations variances
are homogenous or not.

7. ANOVA
• The ‘Analysis of Variance’ (ANOVA) is the appropriate statistical
technique to be used in situations where we have to compare more
than two groups
OR
• It is a powerful statistical procedure for determining if differences in
means are significant and for dividing the variance into components.

8. • Variance (2
) is an absolute measure of dispersion of raw scores around the
sample (group) mean, the dispersion of the scores resulting from their varying
differences (error terms) from the means.
• Mean square – The measure of variability used in the analysis of variance is
called a mean square
• Sum of squared deviation from mean divided by degrees of freedom.
•
• Mean square = Sum of squared deviation from mean
• ----------------------------------------------
• Degrees of freedom

9. • Assumptions in analysis of variance
• The samples are independently drawn
• The population are normally distributed, with common variance
• They occur at random and independent of each other in the groups
• The effects of various components are additive.

10. • Technique for analysis of variance
• One - way ANOVA : Here a single independent variable is involved
• Eg: Effect of pesticide (independent variable) on the oxygen
consumption (dependent variable) in a sample of insect.
• Two -way ANOVA: Here two independent variables are involved.
• Eg: Effects of different levels of combination of a pesticide
(independent variable) and an insect hormone (independent variable)
on the oxygen consumption of a sample of insect.

11. • Working Procedure
• The procedure of calculation in direct method are lengthy as well as time
consuming and this is not popular in practice for all purposes.
• Therefore a short cut method based on the sum of the squares of the
individual values are usually used.
• This method is more convenient.

23. Critical Value = 3.89
F = 22.59

24. Thank you