If the correlation between height and weight of a large...

If the correlation between height and weight of a large group of people is $0.67$, find the $\mathrm{co}$ efficient of determination (as a percentage) and explain what it means. Assume that height is the predictor and weight is the response, and assume that the association between height and weight is linear.

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Key Concepts

Correlation

Correlation is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 imply a strong positive association, values close to -1 indicate a strong negative association, and values near 0 suggest little to no linear relationship.

Coefficient of Determination

The coefficient of determination, often denoted as r², is derived by squaring the correlation coefficient. It represents the proportion of the variance in the response variable that is explained by the predictor variable in a linear regression model. When expressed as a percentage, it shows how much of the variability in the outcome can be accounted for by the predictor.

Linear Regression and Variable Roles

In linear regression, one variable is designated as the predictor (or independent variable) and the other as the response (or dependent variable). This framework is used to model and predict the response based on the predictor. The analysis assumes a linear relationship between the two, making both the correlation and the coefficient of determination central to quantifying and interpreting the association.

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