Geneticists are interested in determining whether a population is in Hardy–Weinberg equilibrium. For a particular gene with two alleles, long‑term data indicate that the expected genotype distribution in the population is 49% homozygous dominant, 42% heterozygous, and 9% homozygous recessive.

A sample of 1000 individuals is collected from the population. Of these, 520 are homozygous dominant, 410 are heterozygous, and 70 are homozygous recessive.

Using a chi‑square goodness‑of‑fit test, determine whether there is evidence that the observed genotype distribution differs from the expected distribution. Calculate the value of the chi‑square test statistic.

Question

Geneticists are interested in determining whether a population is in Hardy–Weinberg equilibrium. For a particular gene with two alleles, long‑term data indicate that the expected genotype distribution in the population is 49% homozygous dominant, 42% heterozygous, and 9% homozygous recessive.

A sample of 1000 individuals is collected from the population. Of these, 520 are homozygous dominant, 410 are heterozygous, and 70 are homozygous recessive.

Using a chi‑square goodness‑of‑fit test, determine whether there is evidence that the observed genotype distribution differs from the expected distribution. Calculate the value of the chi‑square test statistic.

BlackTom AI · Accepted Answer

Chi-square goodness-of-fit statistic is computed as Σ((O − E)^2 / E), so begin by converting the expected percentages to expected counts for the sample of 1000.

1. Relevant knowledge points and concepts
- The chi-square goodness-of-fit test compares observed counts (O) to expected counts (E) under a specified distribution. 
- For each category, the contribution is (O − E)^2 / E, and the test statistic is the sum of these contributions.
- Expected counts = expected proportion × sample size. Degrees of freedom would be k − 1 (here 3 − 1 = 2) if performing hypothesis testing, but the question asks for the test statistic value.

2. Complete reasoning process step by step
- Given expected genotype distribution: 49% homozygous dominant, 42% heterozygous, 9% homozygous recessive.
- Sample size = 1000, observed counts: O1 = 520, O2 = 410, O3 = 70.
- Compute expected counts: E1 = 0.49 × 1000 = 490, E2 = 0.42 × 1000 = 420, E3 = 0.09 × 1000 = 90.

3. Calculations with intermediate steps
- Category 1 (homozygous dominant): O1 − E1 = 520 − 490 = 30; contribution = 30^2 / 490 = 900 / 490 = 1.8367346939.
- Category 2 (heterozygous): O2 − E2 = 410 − 420 = −10; contribution = (−10)^2 / 420 = 100 / 420 = 0.2380952381.
- Category 3 (homozygous recessive): O3 − E3 = 70 − 90 = −20; contribution = 400 / 90 = 4.4444444444.
- Sum the contributions: 1.8367346939 + 0.2380952381 + 4.4444444444 = 6.5192743764 ≈ 6.5193.

4. Why the correct answer is correct
- The calculation follows the chi-square formula exactly, using the provided expected proportions and observed counts. Each term reflects the squared deviation scaled by expected frequency; summing them yields the correct test statistic.

5. Why other common approaches would be incorrect
- Using proportions directly without converting to counts, or using an incorrect expected count (e.g., rounding expected counts before computing contributions) would change the terms and produce an incorrect χ^2. Also, omitting one category or miscomputing (O − E) would lead to wrong results.

6. Conclusion and confirmation
- The chi-square test statistic is therefore 6.519274... which rounded to four decimal places is 6.5193, matching the provided answer 6.5193.

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