Question

You are given Mendel's original data on the inheritance of yellow versus green cotyledons. In his experiment, 8023 individuals were scored. Of the 8023,6022 were yellow and 2001 were green. We expect that yellow and green will appear in a \(3: 1\) ratio respectively. Using the chi-square test \(\left(\mathrm{X}^{2}\right)\), how well does Mendel's data fit our proposed ratio?

Short answer

The chi-square (X²) value for Mendel's data on yellow and green cotyledons is 1.493 with 1 degree of freedom. Comparing this value to the critical value of 3.841 at a 0.05 significance level, we can conclude that Mendel's data fits the proposed 3:1 ratio quite well, as there is no significant difference between the observed and expected ratios.

Step-by-step solution

Calculate Expected Numbers

First, let's find the total number of expected yellow and green cotyledons based on a 3:1 ratio. To do this, we'll divide Mendel's total count (8023) by the sum of the ratio values (3+1=4). Fraction for yellow: \(\frac{3}{4}\) and fraction for green: \(\frac{1}{4}\). Expected yellow cotyledons: \( \frac{3}{4} \times 8023 = 6017.25\) Expected green cotyledons: \( \frac{1}{4} \times 8023 = 2005.75\)

Compute Chi-Square (X²) Value

Next, we'll use the observed and expected numbers to compute the chi-square (X²) value using the formula: \( X^2 = \sum \frac{(O_i - E_i)^2}{E_i} \) Where: \(X^2\) = Chi-Square value \(O_i\) = Observed value \(E_i\) = Expected value For yellow cotyledons: \(X^2_{yellow} = \frac{(6022 - 6017.25)^2}{6017.25} = 0.373\) For green cotyledons: \(X^2_{green} = \frac{(2001 - 2005.75)^2}{2005.75} = 1.120\) Total chi-square (X²) value: \(X^2_{total} = X^2_{yellow} + X^2_{green} = 0.373 + 1.120 = 1.493\)

Determine Degrees of Freedom and Significance Level

Degrees of freedom (df) = Number of categories (yellow and green) - 1 = 2 - 1 = 1 Now, we need to compare the chi-square (X²) value with a critical value from the chi-square distribution table for one degree of freedom and a significance level (p-value). Typically, a significance level of 0.05 (5%) is used as a threshold for accepting or rejecting the null hypothesis. Looking at a chi-square distribution table for df = 1 and p = 0.05, we find a critical value of 3.841. Since our calculated chi-square value (1.493) is less than the critical value (3.841), we can conclude that Mendel's data fits the proposed 3:1 ratio quite well, and there is no significant difference between the observed and expected ratios.

Key concepts

Mendelian Inheritance

Mendelian inheritance is a set of principles regarding how traits are passed down from one generation to the next. These principles were first established by Gregor Mendel in the 19th century through his experiments with pea plants. Mendel discovered that certain traits follow specific patterns of inheritance, which now serve as the foundation of classical genetics.
Mendel's observations led him to formulate two main laws:

• Law of Segregation: Each individual carries two alleles for each trait, and these alleles segregate (separate) during the formation of gametes, ensuring each gamete carries just one allele per trait.
• Law of Independent Assortment: Genes for different traits are inherited independently of each other, given they are on separate chromosomes or far apart on the same chromosome, allowing for various trait combinations.

In the context of Mendel’s experiment on pea plants, the inheritance pattern of yellow versus green cotyledons exemplifies how these traits adhere to Mendelian inheritance, with the dominant yellow trait masking the recessive green trait in a classic 3:1 ratio when observed in a population.

3:1 Ratio in Genetic Inheritance

The 3:1 ratio is a hallmark of Mendelian inheritance for monohybrid crosses. This occurs when a dominant and a recessive allele are paired in a heterozygous organism. The 3:1 phenotypic ratio emerges in the offspring of such a cross, where three of the offspring show the dominant trait and one shows the recessive trait.
This ratio is derived from the fact that when two heterozygous parents (e.g., Yy for yellow cotyledons and yy for green) are crossed, the offspring can have one of the following combinations:

• YY - Homozygous dominant
• Yy - Heterozygous dominant
• Yy - Heterozygous dominant
• yy - Homozygous recessive

This results in three individuals exhibiting the dominant phenotype (yellow) and one exhibiting the recessive phenotype (green). Hence, the 3:1 ratio is evident. It is essential in predicting how traits are likely to occur in subsequent generations during crossbreeding experiments and can be confirmed statistically using tools like the chi-square test.

Degrees of Freedom in Statistical Testing

Degrees of freedom are crucial in statistical testing as they represent the number of independent values that can vary in a calculation involving statistical measures. In the context of a chi-square test, degrees of freedom are determined by the number of categories under consideration minus one.
For example, when analyzing Mendel's experiment with yellow and green cotyledons, there are two categories (yellow and green). Therefore, the degrees of freedom are calculated as:\[ \text{df} = \text{Number of categories} - 1 = 2 - 1 = 1 \]
This number is vital because it helps in determining the critical value from a chi-square distribution table, used to decide if the observed data significantly deviates from the expected values. The number of degrees of freedom influences the shape of the chi-square distribution and thereby affects the chi-square values critical for hypothesis testing. In Mendel's data analysis, this concept allows us to compare the calculated chi-square value against the critical value to ascertain how well the empirical data align with theoretical predictions.