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There is no relationship between correla...

There is no relationship between correlation coefficient and regression coefficient.

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To determine whether the statement "there is no relationship between correlation coefficient and regression coefficient" is true or false, we can analyze the relationship between these two statistical measures step by step. ### Step-by-Step Solution: 1. **Understanding Correlation Coefficient (r)**: - The correlation coefficient (denoted as r) measures the strength and direction of a linear relationship between two variables, X and Y. It ranges from -1 to 1. 2. **Understanding Regression Coefficients (bxy and byx)**: - The regression coefficients are used in regression analysis to predict the value of one variable based on another. - The regression coefficient of Y on X (byx) indicates how much Y changes for a unit change in X. - The regression coefficient of X on Y (bxy) indicates how much X changes for a unit change in Y. 3. **Formulas for Regression Coefficients**: - The formula for the regression coefficient of Y on X (byx) is given by: \[ byx = r \cdot \frac{\sigma_y}{\sigma_x} \] - The formula for the regression coefficient of X on Y (bxy) is given by: \[ bxy = r \cdot \frac{\sigma_x}{\sigma_y} \] 4. **Relationship Between Correlation and Regression Coefficients**: - If we multiply the two regression coefficients, we get: \[ byx \cdot bxy = \left(r \cdot \frac{\sigma_y}{\sigma_x}\right) \cdot \left(r \cdot \frac{\sigma_x}{\sigma_y}\right) \] - Simplifying this, we find: \[ byx \cdot bxy = r^2 \] - Therefore, we can express the correlation coefficient as: \[ r = \sqrt{byx \cdot bxy} \] - This shows that the correlation coefficient is indeed related to the regression coefficients. 5. **Conclusion**: - Since we have established a mathematical relationship between the correlation coefficient and the regression coefficients, the statement "there is no relationship between correlation coefficient and regression coefficient" is **false**. ### Final Answer: The statement is **false**. ---
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