Interpreting the Coefficient of Determination
In Exercis...

Question

Interpreting the Coefficient of Determination

In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Times of Taxi Rides and Fares r = 0.953 (x = time in minutes, y = fare in dollars)

Accepted Answer

Welcome back, everyone. In this problem, a study analyzes the relationship between the number of study OS X and the test scores of students Y. The data yield a linear correlation coefficient of R equals 0.867. Use the given value of R to find the coefficient of determination and the percentage of the total variation in test scores that can be explained by the number of study orders. A says R squared is 0.9482, which means 94.82% of the variation can be explained. B says it's 0.9172, which means 91.72% of the variation can be explained. C says it's 0.8670, which means 86.70% of the variation can be explained while D says it's 0.7517, which means 75.17% of the variation can be explained. Now, how can we find the coefficient of determination and the percentage of the total variation of test scores that can be explained by the number of study hours? will recall that the coefficient of the determination which is denoted as r squared represents the proportion of the variance in the dependent variable Y that is explained by the independent variable X in a linear regression model. So mathematically it's really the square of the linear correlation coefficient. We know that our linear correlation coefficient is equal to 0.867, so the coefficient of determination is going to be 0.867 squared, which is equal to 0.7517. This value helps us determine how well the independent variable explains the variation in the dependent variable that is the study orders of the test scores. This coefficient of determination if we were to rewrite it as a percentage would be 75.17%, and this means that 75.17% of the total variation in test scores then can be explained by the number of study orders. The remaining 24.83% of the variation, OK, so the remaining variation. Could be due to other factors not included in the in the model, OK. Such as the students' prior knowledge, possibly study their study method used, the test difficulty or environmental factors. So the remaining variation is due, let me write that here to other factors, OK. Not included. In this model. Therefore, that tells us then that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Coefficient of Determination (R²)

Linear Correlation Coefficient (r)

Total Variation

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