A variety of pea plant called Blue Persian produces a tall plant with blue seeds. A second variety of pea plant called Spanish Dwarf produces a short plant with white seed. The two varieties are crossed, and the resulting seeds are collected. All of the seeds are white; and when planted, they produce all tall plants. These tall F₁ plants are allowed to self-fertilize. The results for seed color and plant stature in the F₂ generation are as follows:
   F₂ Plant Phenotype    Number
   Blue seed, tall plant.     97
   White seed, tall plant    270
   Blue seed, short plant    33
   White seed, short plant  100
  TOTAL                500


Examine the data in the table by the chi-square test and determine whether they conform to expectations of the hypothesis.

Verified step by step guidance

1. Step 1: Define the hypothesis. Based on the problem, the hypothesis is that the traits (seed color and plant stature) follow Mendelian inheritance patterns. Specifically, seed color and plant stature are controlled by two independent genes, each with two alleles, and the F2 generation should follow a 9:3:3:1 phenotypic ratio.
2. Step 2: Calculate the expected numbers for each phenotype. Using the total number of F2 plants (500) and the expected 9:3:3:1 ratio, calculate the expected number of plants for each phenotype. For example, the expected number for the 'blue seed, tall plant' phenotype is (9/16) * 500. Repeat this calculation for the other three phenotypes.
3. Step 3: Apply the chi-square formula. For each phenotype, calculate the chi-square value using the formula: χ² = Σ((observed - expected)² / expected). Subtract the expected number from the observed number for each phenotype, square the result, divide by the expected number, and sum these values for all phenotypes.
4. Step 4: Determine the degrees of freedom. The degrees of freedom (df) for a chi-square test is calculated as the number of phenotypic categories minus 1. In this case, there are 4 phenotypic categories, so df = 4 - 1 = 3.
5. Step 5: Compare the chi-square value to the critical value. Using a chi-square distribution table, find the critical value for df = 3 at a significance level (e.g., 0.05). Compare the calculated chi-square value to the critical value to determine whether to accept or reject the hypothesis. If the chi-square value is less than the critical value, the data conform to the hypothesis; otherwise, they do not.

Key Concepts

Mendelian Genetics

Mendelian genetics is the study of how traits are inherited through generations based on the principles established by Gregor Mendel. It involves understanding dominant and recessive alleles, where dominant traits mask the expression of recessive ones. In this scenario, the tall plant trait is likely dominant over the short plant trait, and the white seed trait is dominant over the blue seed trait, which is crucial for analyzing the F₂ generation results.

Phenotypic Ratios

Phenotypic ratios represent the relative frequencies of different observable traits in the offspring of a genetic cross. In this case, the F₂ generation's phenotypic ratio can be analyzed to determine if it aligns with expected Mendelian ratios, such as the 9:3:3:1 ratio for a dihybrid cross. Understanding these ratios helps in evaluating whether the observed data supports the initial hypothesis about inheritance patterns.

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant difference between observed and expected frequencies in categorical data. In this context, it will help assess whether the distribution of phenotypes in the F₂ generation deviates from the expected Mendelian ratios. A low chi-square value indicates that the observed data fits the expected model well, while a high value suggests a significant difference, prompting further investigation.