Which statistical test is used to determine whether the observed progeny of a cross fits or differs from the expected values? Name the test and briefly describe the steps involved.

The Chi-Square Test: A Statistical Tool for Genetic Analysis

The Chi-square test is a non-parametric test used to determine if there is a statistically significant association between two categorical variables. In the context of genetics, these variables are the observed and expected frequencies of different phenotypes resulting from a genetic cross.

Steps Involved in Performing the Chi-Square Test

1. Formulate Null and Alternative Hypotheses:
   • Null Hypothesis (H₀): There is no significant difference between the observed and expected frequencies. Any deviation is due to chance.
   • Alternative Hypothesis (H₁): There is a significant difference between the observed and expected frequencies.
2. Create a Table of Observed and Expected Values:

   Organize the data into a table with columns for observed (O) and expected (E) values for each phenotype. The expected values are calculated based on the Mendelian ratios predicted by the cross.

   For example, in a monohybrid cross with a predicted 3:1 phenotypic ratio, if the total number of progeny is 100, the expected values would be 75 (for the dominant phenotype) and 25 (for the recessive phenotype).

3. Calculate the Chi-Square Statistic (χ²):

   The Chi-square statistic is calculated using the following formula:

   χ² = Σ [(O - E)² / E]

   Where:

   • χ² is the Chi-square statistic
   • O is the observed frequency
   • E is the expected frequency
   • Σ represents the sum across all categories
4. Determine the Degrees of Freedom (df):

   The degrees of freedom are calculated as:

   df = (number of categories - 1)

   For example, in a monohybrid cross with two phenotypes, df = 2 - 1 = 1.

5. Determine the Critical Value and p-value:

   Using the calculated χ² value and the degrees of freedom, compare the χ² value to a critical value from a Chi-square distribution table. Alternatively, calculate the p-value, which represents the probability of obtaining the observed results (or more extreme results) if the null hypothesis is true.

6. Interpret the Results:
   • If the calculated χ² value is greater than the critical value, or if the p-value is less than the significance level (typically 0.05), the null hypothesis is rejected. This indicates a statistically significant difference between the observed and expected frequencies.
   • If the calculated χ² value is less than the critical value, or if the p-value is greater than the significance level, the null hypothesis is not rejected. This suggests that the observed deviations are likely due to chance.

Example: Testing a Monohybrid Cross

Suppose a plant breeder performs a monohybrid cross and observes 60 plants with the dominant phenotype and 40 plants with the recessive phenotype. The expected ratio is 3:1. Let's perform the Chi-square test:

Phenotype	Observed (O)	Expected (E)	(O-E)	(O-E)²	(O-E)²/E
Dominant	60	75	-15	225	3.0
Recessive	40	25	15	225	9.0
Total	100	100			12.0

χ² = 12.0, df = 1. The critical value for df=1 and α=0.05 is 3.841. Since 12.0 > 3.841, we reject the null hypothesis, suggesting a significant deviation from the expected 3:1 ratio.