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 \(3.84 .\)

Short answer

(C) No, the \(\chi^2\) value is greater than \(3.84\).

Step-by-step solution

Calculate the expected number of yellow and green plants

Based on the given ratio (3 yellow to 1 green), there are 4 parts in total. Divide 100 plants by 4, and then multiply by the number of yellow and green parts to find the expected counts : Expected yellow plants = \(\frac{3}{4} \times 100 = 75\) , Expected green plants = \(\frac{1}{4} \times 100 = 25\)

Calculate the difference between observed and expected counts

Subtract the expected counts from the observed counts : Difference in yellow plants = Observed yellow plants - Expected yellow plants = \(84 - 75 = 9\), Difference in green plants = Observed green plants - Expected green plants = \(16 - 25 = -9\)

Calculate the chi-square (\(\chi^2\)) value

To find the \(\chi^2\) value, square the differences obtained in Step 2 and then divide by the expected counts, and sum up the results : \(\chi^2 = \frac{(Difference\ in\ yellow\ plants)^2}{Expected\ yellow\ plants} + \frac{(Difference\ in\ green\ plants)^2}{Expected\ green\ plants} = \frac{9^2}{75} + \frac{(-9)^2}{25} = \frac{81}{75} + \frac{81}{25} = 1.08 + 3.24 = 4.32\)

Compare the chi-square value to the critical value

Since the critical value for 1 degree of freedom at a 0.05 significance level is 3.84, we can compare our \(\chi^2\) value to this critical value: \(\chi^2 = 4.32\) , Critical value = \(3.84\) Since the \(\chi^2\) value (4.32) is greater than the critical value (3.84), the experimental data do not match the expected data. Therefore, the correct answer is: \( \boxed{\text{(C) No, the } \chi^2 \text{ value is greater than } 3.84 } \)

Key concepts

Genetics

Genetics is the branch of biology that deals with the study of heredity and variations. Heredity is the process by which parents pass on traits to their offspring through genes. Genes, which are made up of DNA, act as instructions to make molecules called proteins. These genetic instructions are inherited through reproduction and are responsible for the development of traits such as eye color, height, and even susceptibility to certain diseases.

In the example given, genetics plays a central role as it is concerned with the phenotypic outcomes (yellow and green plants) of a genetic cross. A phenotype is the physical expression or characteristics of an organism, which in this case is the color of the pea plants. The phenotype is influenced by the genotype, which is the genetic makeup of the organism, as well as the organism's environment. Understanding how genetics works helps to predict the likelihood of certain traits being passed on to the next generation.

Mendelian Inheritance

Mendelian inheritance is named after Gregor Mendel, who is considered the father of genetics. His studies on pea plants laid the foundation for our understanding of how traits are inherited from one generation to the next. Mendel's first law, the Law of Segregation, states that the two copies of each hereditary factor segregate so that offspring acquire one factor from each parent.

In the context of the given problem, Mendelian inheritance can explain the expected 3:1 ratio of yellow to green plants. Yellow and green are traits determined by a single gene with two alleles. The allele for yellow might be dominant, while the allele for green could be recessive. A 3:1 ratio is indicative of a monohybrid cross, where each parent is heterozygous for the trait (having both a yellow and a green allele). The predicted ratio comes from the possible combinations of alleles each offspring could inherit.

Statistical Significance

Statistical significance is a term used to determine whether the result of a hypothesis test is likely due to something other than mere random chance. In genetics, a chi-square test can be used to determine if the observed phenotypic ratios significantly differ from the expected ratios. A chi-square test provides a way to objectively quantify this difference.

The critical value in a chi-square test is typically determined by the degrees of freedom associated with the test and the accepted level of significance, often set at 0.05 for biological studies. If the chi-square value is greater than the critical value at a specific degree of freedom, the null hypothesis, which postulates no significant difference between observed and expected values, is rejected. As seen in the provided exercise, the value of 4.32 exceeds the critical threshold of 3.84, leading to the conclusion that the data significantly deviates from what was expected under Mendelian inheritance patterns, which may suggest an alternative genetic mechanism, sampling error, or other influencing factors are at work.