Question

Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this toll you about the explained varasion of the data atout the regression ine? About the unexplaned variation? r=0.421

Answer 1

The coefficient of determination, obtained by squaring the correlation coefficient r = 0.421, is approximately 17.72%, indicating that about 17.72% of the variability in the dependent variable is explained by the regression line.

Step-by-step explanation:

The value of the linear correlation coefficient, r, is given as 0.421. To calculate the coefficient of determination, r², we square the correlation coefficient: r² = (0.421)² = 0.177241. The coefficient of determination represents the percentage of variation in the dependent variable that can be explained by the independent variable in the regression model. Therefore, approximately 17.72% of the variation in the dependent variable can be explained by the regression line, leaving the remaining 82.28% of the variation unexplained, which might be due to other factors or random variability.

Answer 2

To calculate the coefficient of determination (r²), you square the correlation coefficient (r). For r = 0.421, r² = 0.1772, meaning that about 17.72% of the variation is explained by the regression line. The remaining 82.28% of the variation is unexplained, indicating other factors might be in play.

Step-by-step explanation:

The question asks how to calculate the coefficient of determination from the linear correlation coefficient and what this statistic tells us about the explained and unexplained variation in the data relative to the regression line. The linear correlation coefficient given is r = 0.421.

To calculate the coefficient of determination, denoted as r², you square the correlation coefficient: r² = 0.421² = 0.1772, or about 17.72% when expressed as a percentage.

The coefficient of determination represents the proportion of the variance in the dependent variable that is predictable from the independent variable. In this case, approximately 17.72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x) using the regression line. Conversely, approximately 82.28% of the variation is unexplained, which indicates that there may be other factors affecting the dependent variable that are not captured by this simple linear model.