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)

검증된 단계별 안내

Step 1: Understand the coefficient of determination (denoted as R²). It represents the proportion of the total variation in the dependent variable (y) that can be explained by the independent variable (x) through the linear relationship.

Step 2: To calculate R², square the given linear correlation coefficient r. In this case, r = 0.953, so compute R² = (0.953)².

Step 3: Once R² is calculated, interpret it as a percentage by multiplying the result by 100. This percentage represents the proportion of the total variation in the fares (y) that can be explained by the time of taxi rides (x).

Step 4: The remaining percentage (100% - R²%) represents the variation in fares that cannot be explained by the time of taxi rides and may be due to other factors.

Step 5: Summarize the interpretation: A high R² value close to 1 indicates a strong linear relationship between the variables, meaning most of the variation in fares can be explained by the time of taxi rides.

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주요 개념

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Coefficient of Determination (R²)

The coefficient of determination, denoted as R², quantifies the proportion of variance in the dependent variable that can be explained by the independent variable in a regression model. It ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect explanation. In this context, R² is calculated as the square of the correlation coefficient (r), providing insight into the strength of the linear relationship between the two variables.

Linear Correlation Coefficient (r)

The linear correlation coefficient, represented as r, measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values around 0 suggest no linear correlation. In the given example, r = 0.953 indicates a very strong positive correlation between taxi ride times and fares.

Total Variation

Total variation refers to the overall variability in the dependent variable, which in this case is the fare of taxi rides. It is the sum of the explained variation (due to the linear relationship with the independent variable) and the unexplained variation (due to other factors). Understanding total variation is crucial for interpreting R², as it helps to contextualize how much of the variability in fares can be attributed to the time of the taxi rides.