Statistical Methods

Unit 7 - Chi-square (X²) distribution

7.2.Test for fixed-ratio hypothesis
Many investigations are carried out to verify empirically some biological phenomena that are expected to occur under some given assumptions. In common carps normal pigmentation is due to a dominant gene B — and its recessive allele bb produces blue pigmentation. F₁ generation will have all individuals (Bb) with normal pigmentation. When F₁’s are crossed to obtain F₂ generation there will be common carps with normal and blue pigmentation in the ratio of 3: 1. Whether this hypothesis of 3: 1 ratio is substantiated by the actual observed data can be ascertained by - test. This test can be applied to test any fixed ratio hypothesis provided the expected ratio is specified before the investigation commences.
If O_i refers to observed frequency and E_i refers to the expected frequency based on the expected ratio hypothesis, then is computed as follows:

Where n is the total number of observations and k is the number of classes. The in (1) has k-1 degrees of freedom. In this test the expected frequency of each class should be more than 5. If any such frequency is small adjacent classes may be grouped, so that the expected frequency is more than 5.
If the calculated value of is greater than the table value of with (k-1) df, at specified level of significance the null hypothesis of specified ratio is rejected.

Example : A sample of 500 fish observed for determining the sex ratio, indicated that 230 were male and 270 female. Do the observed data fit the expected ratio of 1: 1?
(i) Hypothesis
H_o :The observed data fit he ratio 1:1
H₁ :The observed data does not fit the ratio 1:1.
(ii) Test statistic
On the basis of this hypothesis of 1:1 ratio, 250 fish are expected in male and female classes, is calculated as follows:

Sex	Frequency		Oi2	Oi2/Ei
	Observed (Oi)	Expected (Ei)
Male	230	250	52,900	211.60
Female	270	250	72,900	291.60
Total	500	500		503.20

(iii) Statistical decision
The table value of with 1 df at 5% level Significance is 3.841. As computed is less than the table value of , the hypothesis is not rejected.