Question

To assess Mendel's law of segregation using tomatoes, a truebreeding tall variety (SS) is crossed with a true-breeding short variety \((s s) .\) The heterozygous \(F_{1}\) tall plants \((S s)\) were crossed to produce two sets of \(\mathrm{F}_{2}\) data, as follows. \(\begin{array}{cc}\text { Set I } & \text { Set II } \\ 30 \text { tall } & 300 \text { tall } \\ 5 \text { short } & 50 \text { short }\end{array}\) (a) Using the \(\chi^{2}\) test, analyze the results for both datasets. Calculate \(\chi^{2}\) values and estimate the \(p\) values in both cases. (b) From the above analysis, what can you conclude about the importance of generating large datasets in experimental conditions?

Short answer

Answer: Generating large datasets in experimental conditions is essential as it enhances the validity and reliability of conclusions derived from the data. In the analysis of the two plant datasets, the smaller dataset (Set I) did not allow us to conclusively determine if the observed values followed Mendel's 3:1 ratio or not, whereas the larger dataset (Set II) helped us reject the null hypothesis and conclude that the data did not follow Mendel's 3:1 ratio.

Step-by-step solution

Set up Hypothesis and Observed values

To assess Mendel's law of segregation, we need to set up our hypotheses. Null hypothesis (H0): The data follows Mendel's 3:1 ratio for tall and short plants in F2 generation. Alternative hypothesis (H1): The data does not follow Mendel's 3:1 ratio for tall and short plants in F2 generation. Let's list the observed values for both datasets: Set I: 30 tall, 5 short Set II: 300 tall, 50 short

Calculate Expected values

To find the expected values, we will apply Mendel's 3:1 ratio for tall and short plants in F2 generation. For Set I: Total plants = 30 + 5 = 35 Expected ratio: 3:1 -> 3/4 tall, 1/4 short Expected values: \(\frac{3}{4} \times 35 = 26.25\) tall, \(\frac{1}{4} \times 35 = 8.75\) short For Set II: Total plants = 300 + 50 = 350 Expected ratio: 3:1 -> 3/4 tall, 1/4 short Expected values: \(\frac{3}{4} \times 350 = 262.5\) tall, \(\frac{1}{4} \times 350 = 87.5\) short

Calculate Chi-Square values

We can now calculate the \(\chi^2\) values for both datasets using the formula: \(\chi^2 = \sum \frac{(O - E)^2}{E}\) where O = Observed values, E = Expected values For Set I: \(\chi^2 = \frac{(30 - 26.25)^2}{26.25} + \frac{(5 - 8.75)^2}{8.75} = 0.5714 + 1.6071 = 2.1785\) For Set II: \(\chi^2= \frac{(300 - 262.5)^2}{262.5} + \frac{(50 - 87.5)^2}{87.5} = 5.3529 + 16.0714 = 21.4243\)

Find p-values

To find the p-values, we will use the chi-square distribution table with 1 degree of freedom (df = 1). For Set I: \(\chi^{2}= 2.1785\) lies between 2.706 and 0.455 (corresponding p-values 0.10 and 0.50). We can estimate the p-value lies between 0.10 and 0.50. For Set II: \(\chi^{2}= 21.4243\) is greater than 10.828 (corresponding p-value 0.001). Thus, we can estimate the p-value is less than 0.001.

Conclusion

(a) For the results of both datasets, we found the following: Set I: \(\chi^{2}= 2.1785\) and 0.10 < p-value < 0.50 Set II: \(\chi^{2}= 21.4243\) and p-value < 0.001 (b) From the analysis above, we observed that with a larger dataset (Set II), we were able to reject the null hypothesis, indicating the data doesn't follow Mendel's 3:1 ratio. In contrast, with a smaller dataset (Set I), we couldn't conclude if the observed values followed Mendel's 3:1 ratio or not. This highlights the importance of generating large datasets in experimental conditions to enhance the validity and reliability of conclusions derived from the data.

Key concepts

Chi-square (χ^2) test

The Chi-square test is a statistical method used to compare the observed data with the data we would expect to obtain according to a specific hypothesis. In the context of genetics, it is often used to determine whether the observed phenotypic ratios of offspring are in alignment with the expected ratios, presupposed by a genetic model, such as Mendel's laws.

By calculating the Chi-square value and comparing it against a critical value from the Chi-square distribution table, we can make judgments about the data. If the calculated Chi-square value is higher than the table value corresponding to our chosen significance level, we reject the null hypothesis. In the experiment examining Mendel's law of segregation, the Chi-square test helps us to affirm if the observed 3:1 ratio of tall to short plants in the tomatoes' F2 generation effectively matches Mendel's predictions.

Genetics

Genetics is a biological science that focuses on understanding how traits are passed from parents to offspring through genes. The fundamental principles of inheritance were established by Gregor Mendel in the 19th century. Mendel's law of segregation states that during the formation of gametes, the two alleles for a trait separate, so that each gamete receives only one allele. This concept was later understood to be due to the segregation of homologous chromosomes during meiosis. When we examine traits such as plant height in tomatoes, genetics allows us to predict the phenotypic outcomes based on known genotypic ratios and dominance relationships.

It's incredibly important to distinguish between an organism's genotype, its genetic makeup, and its phenotype, the observable physical characteristics. This distinction is at the core of genetic inheritance and is crucial for understanding the predictions made for the tomoto plants' F2 generation in the exercise.

Experimental data analysis

Experimental data analysis is the process of inspecting, cleaning, transforming, and modeling data with the goal of discovering useful information, suggesting conclusions, and supporting decision-making. This process can include various statistical tests, like the Chi-square test, to understand if the relationships within the data are significant or due to random variation.

Within the context of the given exercise, analyzing the data from the tomato cross involves determining whether the observed ratios of phenotypes (tall and short plants) are consistent with the expected 3:1 ratio derived from Mendel's law of segregation. This kind of analysis is not only foundational to validating genetic theories but is also vital in an array of scientific research, ensuring that outcomes are not mistakenly accepted when they occur purely by chance.

Null hypothesis

In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured phenomena. By proposing a null hypothesis, researchers can utilize statistical tests to determine whether there is enough evidence in their data to reject this null hypothesis or not. For the Chi-square test in genetics, the null hypothesis often implies that there is no significant difference between the observed and expected ratios of characteristics.

In our tomato plant experiment, the null hypothesis states that the data follows Mendel’s predicted 3:1 ratio. Only when we have enough evidence that this is unlikely true - generally when the p-value is below a certain threshold - do we reject the null hypothesis in favor of the alternative one, suggesting that other factors might be at play influencing the phenotypic ratios.

Probability values (p-values)

Probability values, commonly known as p-values, are used in statistical hypothesis testing to quantify the evidence against a null hypothesis. The p-value represents the probability of observing the collected data, or something more extreme, assuming the null hypothesis is true. In genetics, a low p-value indicates that the observed genetic ratios are unlikely to have occurred by random chance alone, thus suggesting the potential influence of other genetic factors or errors in assumption.

In our exercise, the p-values for the two datasets of tomato plants suggest different conclusions. For the first set, the p-value did not provide enough evidence to reject the null hypothesis, leaving us uncertain about the adherence to the 3:1 ratio. However, for the second set, the extremely low p-value indicates a strong reason to reject the null hypothesis, suggesting that the observed ratio of tall to short plants differs significantly from the expected Mendelian ratio.