Certainly! Since the image refers to calculations but does **not provide actual numbers**, I'll guide you through the **steps and explanations** would use to answer these questions in a genetics/statistics context. --- ## 13.2 a. Calculate the correlations ### Step 1: Write the Formulae - **Genetic correlation (\( r_G \))** between two traits (A = 50-meter time, B = lifetime winnings): \[ r_G = \frac{\text{Cov}_G(A,B)}{\sqrt{V_{G,A} \cdot V_{G,B}}} \] - Where \( \text{Cov}_G(A,B) \) is the genetic covariance between A and B. - \( V_{G,A} \) and \( V_{G,B} \) are the genetic variances of A and B. **Explanation:** This measures the extent to which genetic factors influencing 50-meter time are correlated with those influencing lifetime winnings. --- ### Step 2: Environmental and Phenotypic Correlations - **Environmental correlation (\( r_E \))**: \[ r_E = \frac{\text{Cov}_E(A,B)}{\sqrt{V_{E,A} \cdot V_{E,B}}} \] - Where \( \text{Cov}_E(A,B) \) is the environmental covariance, and \( V_{E,A} \), \( V_{E,B} \) are the environmental variances. **Explanation:** This shows the association between the environmental factors affecting both traits. - **Phenotypic correlation (\( r_P \))**: \[ r_P = \frac{\text{Cov}_P(A,B)}{\sqrt{V_{P,A} \cdot V_{P,B}}} \] - Where \( \text{Cov}_P(A,B) \) is the total phenotypic covariance, and \( V_{P,A} \), \( V_{P,B} \) are phenotypic variances. **Explanation:** This is the overall observed correlation, due to both genetic and environmental factors. --- ### Final Answer (Summary Table) | Correlation | Formula | Explanation | |--------------------------|------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------| | Genetic correlation | \( r_G = \frac{\text{Cov}_G(A,B)}{\sqrt{V_{G,A} V_{G,B}}} \) | Correlation due to shared genetic effects | | Environmental correlation| \( r_E = \frac{\text{Cov}_E(A,B)}{\sqrt{V_{E,A} V_{E,B}}} \) | Correlation due to shared environmental effects | | Phenotypic correlation | \( r_P = \frac{\text{Cov}_P(A,B)}{\sqrt{V_{P,A} V_{P,B}}} \) | Overall observed correlation (genetic + environmental) | --- ## 13.2 b. Calculate the regressions ### Step 1: Write the Regression Formulae - **Genetic regression (I):** \[ b_{G, \text{winnings|time}} = \frac{\text{Cov}_G(\text{winnings}, \text{time})}{V_{G, \text{time}}} \] **Explanation:** This is the expected change in winnings per unit change in 50-meter time, due to genetic factors. --- ### Step 2: Environmental and Phenotypic Regressions - **Environmental regression (II):** \[ b_{E, \text{winnings|time}} = \frac{\text{Cov}_E(\text{winnings}, \text{time})}{V_{E, \text{time}}} \] **Explanation:** Expected change in winnings per unit change in 50-meter time, due to environmental factors. - **Phenotypic regression (III):** \[ b_{P, \text{winnings|time}} = \frac{\text{Cov}_P(\text{winnings}, \text{time})}{V_{P, \text{time}}} \] **Explanation:** Overall expected change in winnings per unit change in 50-meter time, observed in the population. --- ### Final Answer (Summary Table) | Regression Type | Formula | Explanation | |----------------------|------------------------------------------------------------------------------|-----------------------------------------------------------------------| | Genetic | \( b_G = \frac{\text{Cov}_G(\text{winnings}, \text{time})}{V_{G,\text{time}}} \) | Change in winnings per 1-unit genetic change in 50-m time | | Environmental | \( b_E = \frac{\text{Cov}_E(\text{winnings}, \text{time})}{V_{E,\text{time}}} \) | Change in winnings per 1-unit environmental change in 50-m time | | Phenotypic | \( b_P = \frac{\text{Cov}_P(\text{winnings}, \text{time})}{V_{P,\text{time}}} \) | Change in winnings per 1-unit observed change in 50-m time | --- ## Summary You now have the **formulas and explanations** for all types of correlations and regressions asked. If you have the actual covariance and variance values from Problem 13.1, you could substitute them directly into these formulas for your numerical answers.