Offspring of certain fruit flies may have yellow or ebony bodies and normal wings or short wings. Genetic theory predicts that these traits will appear in the ratio 9:3:3:1 (9 yellow, normal: 3 yellow, short: 3 ebony, normal: 1 ebony, short). A researcher checks 100 randomly selected fruit flies and finds the distribution of the traits to be 59, 20, 11, and 10, respectively. Are the results this researcher observed consistent with the theoretical distribution predicted by the genetic model

Offspring of fruit flies may have yellow or ebony bodies and normal wings or short wings.
Expected phenotypic ratio 9:3:3:1
9 yellow, normal: 3 yellow, short: 3 ebony, normal: 1 ebony, short
Sample size , N = 100
Phenotypic distribution 59:20:11:10

To know if these results are consistent with the expected ones, we need to develop a chi-square analysis. To do it we need the following information

Chi square= ∑ ((Obs-Exp)²/Exp)

- ∑ is the sum of the terms

- Obs are the Observed individuals

- Exp are the Expected individuals

Freedom degrees = K – 1

- K =genotypes number = 4

Significance level, 5% = 0.05

Table value = Critical value

First, define the hypothesis:

Hypothesis: The allele of this population will assort independently. The population is in equilibrium

H₀= Individuals will be equally distributed.

H₁ = Individuals will not be equally distributed.

Second, we need to get the expected number of individuals with each phenotype. To do it, we will use the expected phenotypic ratio and the total number of individuals in the sample. We can just perform a three simple rule, as follows:

16 ------------------------------------ 100 individuals in the sample ------- 100%

9 yellow, normal ---------------- X = 56.25 individuals -------------------56.25%

3 yellow, short ------------------- X = 18.75 individuals --------------------18.75%

3 ebony, normal -----------------X = 18.75 individuals ---------------------18.75%

1 ebony, short ---------------------X = 6.25 individuals ----------------------6.25%

Now that we know the expected numbers of individuals with each genotype, we can compare them with the observed ones.

yellow, normal yellow, short ebony, normal ebony, short

Expected 56.25 18.75 18.75 6.25

Observed 59 20 11 10

The chi-square value = Σ(Obs-Exp)²/Exp.

So now we need to calculate (Obs-Exp)²/Exp

yellow, normal

(Obs-Exp)²/Exp = (59-56.25)²/56.25 = 0.134

yellow, short

(Obs-Exp)²/Exp = (20 - 18.75)²/18.75 = 0.083

ebony, normal

(Obs-Exp)²/Exp = (11 - 18.75 )²/18.75 = 3.203

ebony, short

(Obs-Exp)²/Exp = (10 - 6.25)²/6.25 = 2.25

X² = Σ(Obs-Exp)²/Exp = 0.134 + 0.083 + 3.203 + 2.25 = 5.67

X² = 5.67
Significance level = 0.05
Degrees of freedom = genotypes number - one = 4 - 1 = 3
Critical value or table value = 9.348

P₀.₀₅ > X2

9.348 > 5.67

There is not enough evidence to reject the null hypothesis. The genotypes might be in equilibrium, and there might be an independent assortment.