Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경

Larson 8th Edition

Ch. 9 - Correlation and Regression

문제 9.3.7

9장, 문제 9.3.7

"In Exercises 7-10, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
7. r =0.465"

검증된 단계별 안내

Identify the given correlation coefficient, which is

r = 0.465

Calculate the coefficient of determination by squaring the correlation coefficient:

r^{2} = (0.465)^2

. This value is denoted as

r^2

Interpret

r^2

as the proportion of the total variation in the dependent variable that is explained by the independent variable through the regression line. It represents the explained variation.

Understand that the unexplained variation is the remaining proportion of the total variation, which can be found by subtracting

r^2

from 1, i.e.,

1 - r^2

Summarize that a higher

r^2

value indicates a better fit of the regression line to the data, meaning more variation is explained, while a lower

r^2

means more variation remains unexplained.

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주요 개념

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Correlation Coefficient (r)

The correlation coefficient, denoted 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 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 indicate weak or no linear relationship.

Coefficient of Determination (r²)

The coefficient of determination, r², is the square of the correlation coefficient and represents the proportion of the variance in the dependent variable explained by the independent variable in a regression model. It ranges from 0 to 1, with higher values indicating a better fit of the regression line to the data.

Explained vs. Unexplained Variation

Explained variation refers to the portion of total variation in the data accounted for by the regression model, quantified by r². Unexplained variation is the remaining variation not captured by the model, representing random error or other factors. Together, they sum to the total variation in the dependent variable.

"In Exercises 7-10, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?7. r =0.465"