Question

In 1950 a scientist discovered a population of red and green birds on a small tropical island. They are the same species, but some individuals were red, and some green. The scientist recorded 135 red birds and 65 green birds. The allele for red (R) is dominant over green (r). Before the scientist left, the ship's cats jumped off. They established a cat colony and began hunting the birds. Today you are returning to the island. You want to know if cat depredation has had an effect on the alleles in the bird population. You count 20 red birds and 180 green birds.

1. What is the value of the R phenotype observed?
2. What is the value of the r phenotype observed?
3. What is the value of the R phenotype expected?
4. What is the value of the r phenotype expected?
5. What is your X2 value?
6. what is your p-value?

Answer 1

The value of the R phenotype observed is 20, and the value of the r phenotype observed is 180. The values of the R and r phenotypes expected can be calculated using the Hardy-Weinberg equation. The X^2 value and p-value can be calculated to assess if there has been an effect due to cat depredation.

Step-by-step explanation:

The value of the R phenotype observed is 20, as there are 20 red birds in the population.

The value of the r phenotype observed is 180, as there are 180 green birds in the population.

The value of the R phenotype expected can be calculated using the Hardy-Weinberg equation: p² + 2pq. Since the red allele is dominant, the frequency of the R allele is equal to the proportion of red birds in the population. So, the R phenotype expected can be calculated as: (135/200)² + 2(135/200)(65/200).

The value of the r phenotype expected can be calculated using the Hardy-Weinberg equation: q². Since the green allele is recessive, the frequency of the r allele is equal to the proportion of green birds in the population. So, the r phenotype expected can be calculated as: (65/200)².

X^2 (chi-square) value can be calculated by comparing the observed and expected values for each phenotype (red and green) using the formula: ((Observed - Expected)² / Expected).

p-value can be calculated using the chi-square value and degrees of freedom. Degrees of freedom can be calculated as (number of phenotypes - 1). The p-value represents the probability of obtaining a chi-square value as extreme as the one observed, assuming that there is no significant difference between the observed and expected values.

Answer 2

Sure, let's break this down:

Given:
- Initial count: 135 red birds, 65 green birds
- Current count: 20 red birds, 180 green birds

1. **Value of the R phenotype observed:** 20 red birds
2. **Value of the r phenotype observed:** 180 green birds

To find the expected values, we'll use the Hardy-Weinberg equation:

\[ p^2 + 2pq + q^2 = 1 \]

Where:
- \( p \) represents the frequency of the dominant allele (R)
- \( q \) represents the frequency of the recessive allele (r)
- \( p^2 \) represents the frequency of the homozygous dominant genotype (RR)
- \( q^2 \) represents the frequency of the homozygous recessive genotype (rr)
- \( 2pq \) represents the frequency of the heterozygous genotype (Rr)

Let's calculate the expected values.

Given the initial counts:
- Total initial population = 135 (red) + 65 (green) = 200 birds

Calculating allele frequencies:
- \( p \) (frequency of R allele) = \( \sqrt{135/200} \) ≈ 0.733
- \( q \) (frequency of r allele) = 1 - \( p \) ≈ 0.267

Expected values:
- \( p^2 \) (RR) = \( (0.733)^2 \) ≈ 0.537
- \( q^2 \) (rr) = \( (0.267)^2 \) ≈ 0.071
- \( 2pq \) (Rr) = 2 * 0.733 * 0.267 ≈ 0.391

3. **Value of the R phenotype expected:** \( p^2 \times \) Total count = 0.537 * 200 ≈ 107.4 (round to 107)
4. **Value of the r phenotype expected:** \( q^2 \times \) Total count = 0.071 * 200 ≈ 14.2 (round to 14)

Next, let's calculate the Chi-squared (\( X^2 \)) value:

\[ X^2 = \sum \frac{{(O - E)^2}}{E} \]

Where \( O \) is the observed value and \( E \) is the expected value.

For each phenotype:
- \( X^2 \) for R phenotype: \( \frac{{(20 - 107)^2}}{107} \)
- \( X^2 \) for r phenotype: \( \frac{{(180 - 14)^2}}{14} \)

5. **\( X^2 \) value:** Calculate the sum of both \( X^2 \) values.

Finally, to find the p-value, you'll use a Chi-squared distribution table with one degree of freedom (df = 1) and compare the \( X^2 \) value obtained to determine the significance level.