Problem 6 - AP Biology Premium Prep 2021

Question

Two pea plants are crossed, and a ratio of 3 yellow plants to 1 green plant is expected in the offspring. It is found that out of 100 plants phenotyped, 84 are yellow and 16 are green. Do the experimental data match the expected data? (A) Yes, the \(\chi^{2}\) value is greater than 3.84 . (B) Yes, the \(\chi^{2}\) value is smaller than \(3.84 .\) (C) No, the \(\chi^{2}\) value is greater than 3.84 . (D) No, the \(\chi^{2}\) value is smaller than 3.84 .

Step-by-Step Solution

Answer

(C) No, the χ² value is greater than 3.84.

Step 1: Identify Observed and Expected Frequencies

The problem provides us with the observed frequencies: 84 yellow plants and 16 green plants. The expected ratio is 3 yellow plants to 1 green plant. If we have 100 plants, we expect 3/4 of them to be yellow and 1/4 of them to be green. Thus, the expected frequencies are 75 yellow plants and 25 green plants.

Step 2: Calculate the χ² Value

To calculate the χ² value, we use the formula: \[\chi^2 = \sum \frac{(O-E)^2}{E}\] where \(O\) represents the observed frequency and \(E\) represents the expected frequency. For yellow plants: \[\frac{(84-75)^2}{75} = \frac{(9)^2}{75} = \frac{81}{75}\] For green plants: \[\frac{(16-25)^2}{25} = \frac{(-9)^2}{25} = \frac{81}{25}\] Summing the two values, we find: \[\chi^2 = \frac{81}{75} + \frac{81}{25} = 1.08 + 3.24 = 4.32\]

Step 3: Compare χ² Value to the Critical Value

Now we need to compare the calculated χ² value (4.32) with the critical value (3.84). If the χ² value is greater than the critical value, the experimental data doesn't match the expected data. In this case, the χ² value (4.32) is greater than the critical value (3.84). Therefore, the correct answer is: (C) No, the χ² value is greater than 3.84.

Key Concepts

Mendelian Inheritance

Mendelian inheritance refers to the set of principles related to the transmission of hereditary characteristics from parent organisms to their offspring. These principles were initially discovered by Gregor Mendel in the mid-19th century. Mendel's experiments with pea plants identified how traits are inherited in distinct ratios, starting with his famous monohybrid cross, which showed a 3:1 phenotypic ratio in the second generation. In this context, Mendel demonstrated that traits are governed by pairs of alleles, segregating independently during gamete formation.

Through Mendel's laws, specifically the Law of Segregation and the Law of Independent Assortment, we understand:

• Alleles separate during gamete formation, resulting in gametes carrying only one allele for each trait.
• Different traits are passed on independently of each other due to the random assortment of chromosomes.

The concept of dominant and recessive alleles also emerged, where the dominant allele masks the effect of the recessive allele in heterozygous combinations. Hence, Mendelian inheritance sets the foundation for genetic predictions, such as expecting a 3:1 ratio in offspring phenotypes in monohybrid crosses, as seen in the exercise.

Phenotypic Ratios

Phenotypic ratios describe the expected distribution of observable traits (phenotypes) in offspring. These ratios derive from the genetic composition of the parents and the probabilistic outcomes proposed by Mendel's laws. A classic example of a phenotypic ratio in Mendelian genetics is the 3:1 ratio, which appears in the case of a monohybrid cross between two heterozygous organisms.

In the specific exercise, the phenotypic ratio of 3 yellow to 1 green plant arises from crossing heterozygous pea plants. This means:

• Each parent plant carries one allele for yellow (Y) and one for green (y), making them heterozygous (Yy).
• The combination of alleles in the offspring determines the phenotype, producing a 3:1 ratio of yellow to green plants.

This ratio reflects the probability that any offspring will have a particular phenotype, based on dominant and recessive allele interactions. Phenotypic ratios allow scientists to predict outcomes in genetic crosses, essential for understanding inheritance patterns.

Statistical Analysis in Biology

Statistical analysis plays a crucial role in biology, particularly when examining genetic crosses. One common method is the chi-square (\(\chi^2\)) test, which determines if the observed data significantly deviate from expected data. In biological contexts, this test helps confirm or refute predictions based on genetic models.

Here's a brief overview of how the chi-square test applies to the exercise:

• First, establish the observed data (84 yellow, 16 green) and expected data (75 yellow, 25 green) based on Mendelian predictions.
• Calculate the chi-square value using the formula \[\chi^2 = \sum \frac{(O-E)^2}{E}\], where \(O\) is the observed frequency and \(E\) is the expected frequency.
• Sum the results for both yellow and green plants to get the total \(\chi^2\) value.

In this case, the calculated \(\chi^2\) value of 4.32 exceeds the critical value of 3.84, indicating the observed results differ significantly from expectations, thereby leading to the conclusion that the actual phenotypic outcome does not mirror the expected 3:1 ratio. Understanding these analyses allows biologists to quantitatively evaluate genetic hypotheses and validate theoretical predictions.