Question

In assessing data that fell into two phenotypic classes, a geneticist observed values of \(250: 150 .\) She decided to perform a \(x^{2}\) 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 \(x^{2}\) values for each hypothesis. What can be concluded about each hypothesis?

Short answer

Answer: Based on the calculated chi-square values, Hypothesis B (1:1 ratio) has a slightly better fit for the given data with a chi-square value of 25 compared to Hypothesis A's chi-square value of 33.33. However, it's important to note that both values are quite high, indicating a poor fit for both hypotheses, and, without a critical chi-square value or a p-value, we cannot make definite conclusions on accepting or rejecting these hypotheses.

Step-by-step solution

Calculate the expected values for each hypothesis

First, we'll calculate the expected values for each hypothesis by applying the given ratios and summing both phenotypic classes' occurrences (250 and 150). Total occurrences: 250 + 150 = 400 For Hypothesis A (3:1 ratio), Expected value for Class 1: (3/4) * 400 = 300 Expected value for Class 2: (1/4) * 400 = 100 For Hypothesis B (1:1 ratio), Expected value for Class 1: (1/2) * 400 = 200 Expected value for Class 2: (1/2) * 400 = 200

Calculate the chi-square value for each hypothesis

Now, we'll calculate the chi-square value for each hypothesis using the formula \(x^2 = \sum\frac{(Observed - Expected)^2}{Expected}\). For Hypothesis A (3:1 ratio), \(x^2_A = \frac{(250 - 300)^2}{300} + \frac{(150 - 100)^2}{100}\) \(x^2_A = \frac{(-50)^2}{300} + \frac{50^2}{100}\) \(x^2_A = \frac{2500}{300} + \frac{2500}{100}\) \(x^2_A = 8.33 + 25\) \(x^2_A = 33.33\) For Hypothesis B (1:1 ratio), \(x^2_B = \frac{(250 - 200)^2}{200} + \frac{(150 - 200)^2}{200}\) \(x^2_B = \frac{50^2}{200} + \frac{(-50)^2}{200}\) \(x^2_B = \frac{2500}{200} + \frac{2500}{200}\) \(x^2_B = 12.5 + 12.5\) \(x^2_B = 25\)

Draw conclusions based on chi-square values

Both chi-square values are quite high, which indicates a relatively poor fit for both hypotheses. However, since we are not given a critical chi-square value or a p-value, we cannot determine the level of significance for rejecting or accepting these hypotheses. To conclude, the calculated chi-square values for each hypothesis are: - Hypothesis A (3:1 ratio): \(x^2_A = 33.33\) - Hypothesis B (1:1 ratio): \(x^2_B = 25\) Both hypotheses have high chi-square values, indicating a relatively poor fit, but we cannot make any definite conclusions on the level of significance for accepting or rejecting these hypotheses.

Key concepts

Phenotypic Ratio

The phenotypic ratio is a cornerstone concept in genetics as it represents the distribution of observable traits, or phenotypes, in the offspring of a particular cross. By understanding the phenotypic ratio, researchers can make predictions about the genetic makeup, or genotype, of the organisms involved. In the case of classical Mendelian inheritance, phenotypic ratios like 3:1 or 1:1 are frequently observed in offspring from certain types of genetic crosses, primarily when looking at a single gene with two alleles - one being dominant and the other recessive.

For example, when a geneticist notices a 250:150 phenotypic ratio in the data, it reflects the frequency of two different traits observed in a population. Determining whether this observed ratio fits well with expected Mendelian ratios, such as 3:1 or 1:1, is achieved by using a chi-square analysis. This statistical test will indicate how close the observed data is to the expected phenotypic ratio, giving insight into the genetic mechanisms at play.

Null Hypothesis

In statistical analysis, the null hypothesis is a general statement or default position that there is no relationship between two measured phenomena. In genetics, the null hypothesis typically propounds that the observed data will conform to a certain genetic model or ratio, such as the Mendelian ratios mentioned earlier. In the exercise, the geneticist is making two separate null hypotheses: Hypothesis A suggests that the phenotypic ratio should be 3:1, while Hypothesis B posits a 1:1 ratio.

By setting up these null hypotheses, the researcher is asserting an expectation against which the actual observed data is compared. If the chi-square analysis shows that the data significantly deviates from this expectation, the null hypothesis may be rejected. Crucially, the acceptance or rejection of a null hypothesis depends on the chi-square value relative to a critical value from the chi-square distribution, which is determined by the degrees of freedom and the chosen level of significance (e.g., p-value).

Expected Value

Expected value, in the context of chi-square analysis in genetics, refers to the number of individuals exhibiting a certain phenotype that one would expect to see in a given population if the null hypothesis is true. It's calculated based on the total number of occurrences and the expected phenotypic ratio postulated by the null hypothesis. For instance, in the exercise, we calculate the expected numbers of each class in a 3:1 ratio as 300 and 100, respectively, if the total number of offspring is 400.

The expected value serves as a benchmark against which the observed data is evaluated. When performing a chi-square analysis, the difference between expected and observed values is crucial to understanding whether or not there is a significant discrepancy that could lead to rejection of the null hypothesis. It is this comparison that sheds light on the validity of the proposed genetic model in explaining the distribution of traits.

Genetic Data Analysis

Genetic data analysis encompasses a broad array of statistical techniques used to analyze patterns in genetic data, such as the distribution of phenotypic traits in a population. Chi-square analysis is one of the primary tools used in this field. This non-parametric statistical test is designed to compare the observed values with the expected ones under the null hypothesis. In other words, it answers the question of whether there are significant differences between what was observed in a genetic cross and what would be expected based on a specific genetic model (like the Mendelian ratios).

When conducting a chi-square test, we calculate the chi-square value and compare it with a critical value from a chi-square distribution table. Depending on whether the calculated value exceeds the critical value, the null hypothesis can either be rejected (suggesting that the observed distribution doesn't fit the expected genetic model) or not rejected (indicating compatibility between observed and expected ratios). The exercise doesn't provide a p-value or critical chi-square value, which would be essential to definitively accept or reject the null hypotheses. From the provided data, though, it is clear that there is a discrepancy between the observed and expected values for both hypotheses, signaling a potential need to reevaluate the proposed genetic models.