Question

In assessing data that fell into two phenotypic classes, a geneticist observed values of \(250: 150 .\) She decided to perform a \(\chi^{2}\) analysis by using the following two different null hypotheses: (a) the data fit a 3: 1 ratio, and (b) the data fit a 1: 1 ratio. Calculate the \(\chi^{2}\) values for each hypothesis. What can be concluded about each hypothesis?

Short answer

Answer: Based on the calculated \(\chi^{2}\) values (\(33.333\) for hypothesis (a) and \(25\) for hypothesis (b)), hypothesis (b), the 1:1 ratio, better fits the given data. However, a formal conclusion should be drawn by using \(\chi^{2}\) critical values or p-values, along with the level of significance and degrees of freedom.

Step-by-step solution

Calculate the Expected values for each hypothesis

For hypothesis (a), the expected ratio is 3:1 and for hypothesis (b), the expected ratio is 1:1. We need to calculate the expected values for each phenotype class according to these ratios. First, we find the total number of individuals: \(250 + 150 = 400\). For hypothesis (a), the expected ratio is 3:1, so we have: - Expected value for the first class: \(\frac{3}{(3+1)} \times 400 = 300\) - Expected value for the second class: \(\frac{1}{(3+1)} \times 400 = 100\) For hypothesis (b), the expected ratio is 1:1, so we have: - Expected value for the first class: \(\frac{1}{(1+1)} \times 400 = 200\) - Expected value for the second class: \(\frac{1}{(1+1)} \times 400 = 200\)

Calculate the \(\chi^{2}\) values for each hypothesis

For both hypotheses, we will use the \(\chi^{2}\) formula: \(\chi^{2} = \sum \frac{(O - E)^2}{E}\) Where \(O\) is the observed value, and \(E\) is the expected value. For hypothesis (a): - \(\chi^{2} = \frac{(250-300)^{2}}{300} + \frac{(150-100)^{2}}{100}\) - \(\chi^{2} = \frac{(-50)^{2}}{300} + \frac{50^{2}}{100}\) - \(\chi^{2} = \frac{2500}{300} + \frac{2500}{100} = 8.333 + 25 = 33.333\) For hypothesis (b): - \(\chi^{2} = \frac{(250-200)^{2}}{200} + \frac{(150-200)^{2}}{200}\) - \(\chi^{2} = \frac{50^{2}}{200} + \frac{(-50)^{2}}{200}\) - \(\chi^{2} = \frac{2500}{200} + \frac{2500}{200} = 12.5+12.5 = 25\)

Interpret the results

The calculated \(\chi^{2}\) values are \(33.333\) for hypothesis (a) and \(25\) for hypothesis (b). The lower the \(\chi^2\) value, the better the fit between the observed data and the expected values under a given null hypothesis. Therefore, based on our calculations, hypothesis (b) (i.e., the data fits a 1:1 ratio) is more likely to be true. However, to formally conclude which hypothesis best fits or clearly reject one of them, the analysis should be completed using \(\chi^2\) critical values or p-values, which requires the level of significance (such as \(0.05\) or \(0.01\)), and a degree of freedom (in this case, \(1\)) in order to draw any definite conclusions.

Key concepts

Genetic Ratios

Genetic ratios are fundamental when analyzing inheritance patterns. They reflect the expected distribution of genetic traits based on Mendelian laws. For instance, in a 3:1 ratio, three parts of the population exhibit a dominant trait, while one part exhibits the recessive trait. This particular ratio arises from a classic Mendelian monohybrid cross, where both parents are heterozygous for a given trait.

Understanding genetic ratios helps predict the outcome of genetic crosses in terms of phenotypic classes. It's important to remember that these ratios are based on probability, assuming random mating and independent assortment of alleles. Variations can occur in experimental results due to sampling error, mutation, or other genetic factors.

• A 3:1 ratio is typical in situations where one trait is completely dominant.
• A 1:1 ratio usually indicates a direct segregation of alleles, often found in cases of test crosses.

Grasping these ratios allows scientists to form and test hypotheses about genetic patterns using statistical techniques like the Chi-square test.

Null Hypothesis

The null hypothesis is a starting point in statistical analysis, often used to test whether the differences observed in data are due to chance or a specific cause. In genetics, a null hypothesis might state that there's no significant difference between observed genetic outcomes and expected genetic ratios.

For example, in our exercise, two null hypotheses were considered: one that the data follows a 3:1 ratio, and another that it fits a 1:1 ratio. The purpose of the null hypothesis is to allow researchers to conduct a Chi-square test, which compares observed and expected data to determine if any significant deviations exist.

• If the Chi-square result is high, it implies a significant difference, suggesting the null hypothesis should be rejected.
• If the result is low, it suggests that any observed differences are likely due to random chance, and the null hypothesis cannot be rejected.

This approach is essential for objectively interpreting genetic data and understanding the biological processes behind inheritance patterns.

Phenotypic Classes

Phenotypic classes refer to the various observable traits in a population, classified based on specific characteristics like color, shape, or size. These classes help scientists categorize genetic information easily.

In our genetic exercise, the phenotypic classes were divided into two: first and second, correlating with observed counts of 250 and 150 respectively. Each class corresponds to a different expression of a genetic trait resulting from the combination of alleles inherited from the parents.

• The first phenotypic class often represents the dominant trait.
• The second class usually indicates a recessive trait or alternate phenotype.

Phenotypic classification is crucial for analyzing genetic patterns as it provides the data needed to test hypotheses about genetic ratios and predictions. By observing these classes, geneticists can perform statistical tests to evaluate the validity of predicted genetic models, such as the ratios expected from Mendelian inheritance.