Figure 2.12 shows the results of Mendel's test-cross analysis of independent assortment. In this experiment, he first crossed pure-breeding round, yellow plants to pure-breeding wrinkled, green plants. The round yellow  are crossed to pure-breeding wrinkled, green plants. Use chi-square analysis to show that Mendel's results do not differ significantly from those expected.

Verified step by step guidance

1. Understand the problem: Mendel performed a test cross to analyze independent assortment. The parental generation consisted of pure-breeding round, yellow plants (RRYY) crossed with pure-breeding wrinkled, green plants (rryy). The F1 generation (RrYy) was then test-crossed with wrinkled, green plants (rryy). The goal is to use chi-square analysis to determine if the observed results align with the expected results based on Mendel's laws.
2. Determine the expected phenotypic ratio: According to Mendel's law of independent assortment, the two traits (seed shape and seed color) segregate independently. This means the expected phenotypic ratio for the offspring of the test cross (RrYy × rryy) is 1:1:1:1 for the four phenotypes: round yellow, round green, wrinkled yellow, and wrinkled green.
3. Calculate the expected values: Multiply the total number of offspring by the expected ratio for each phenotype. For example, if the total number of offspring is N, the expected number for each phenotype is N/4. Represent this mathematically as: $\frac{N}{4}$.
4. Perform the chi-square calculation: Use the formula ${\chi }^2=\sum _i^n\frac{{(}^O}{E}$, where O represents the observed number of offspring for each phenotype, E represents the expected number, and the summation is over all phenotypic categories. Calculate the chi-square value by summing the contributions from all four phenotypes.
5. Compare the chi-square value to the critical value: Determine the degrees of freedom (df), which is the number of phenotypic categories minus 1. In this case, df = 4 - 1 = 3. Use a chi-square table to find the critical value at the desired significance level (e.g., 0.05). If the calculated chi-square value is less than the critical value, the observed results do not differ significantly from the expected results, supporting Mendel's hypothesis of independent assortment.

Key Concepts

Mendelian Genetics

Mendelian genetics is the study of how traits are inherited through generations, based on the principles established by Gregor Mendel. His experiments with pea plants led to the formulation of key concepts such as dominant and recessive traits, as well as the laws of segregation and independent assortment, which describe how alleles segregate during gamete formation.

Chi-Square Analysis

Chi-square analysis is a statistical method used to determine whether there is a significant difference between observed and expected frequencies in categorical data. In the context of Mendel's experiments, it helps assess whether the ratio of phenotypes observed in offspring aligns with the expected Mendelian ratios, thus validating or refuting the hypothesis of independent assortment.

Independent Assortment

The principle of independent assortment states that alleles for different traits segregate independently of one another during gamete formation. This means that the inheritance of one trait does not influence the inheritance of another, leading to a variety of genetic combinations in the offspring, which Mendel demonstrated through his dihybrid crosses.