The correlation coefficient between heart weight and body weight in a group of mice isr = 0.765. Using body weight as the Independent Variable we can conclude that:

Question

BlackTom AI · Accepted Answer

The question provides the correlation coefficient r = 0.765 between heart weight and body weight in mice, and asks what can be concluded when using body weight as the independent variable.
Option a: '76.5% of the variation in heart weight is explained by the variation in body weights' This statement confuses r with r^2. The proportion of variation explained by the linear relationship is r^2, not r. Since r = 0.765, r^2 = 0.585 (approximately 58.5%), not 76.5%. So this option is incorrect.
Option b: 'heart weight is 58.5% of body weight' This misinterprets the units and meaning of the relationship. A correlation measures association, not a direct proportion or a ratio of the two variables. It does not imply heart weight equals 0.585 times body weight. This option is incorrect because it misstates the meaning of the correlation and the nature of the relationship.
Option c: 'heart weight is 76.5% of body weight' This again confuses the correlation value with a direct proportional relationship. A correlation coefficient expresses strength and direction of linear association, not a fixed percentage relationship between the two variables. This option is incorrect for the same conceptual reason as option b.
Option d: '76.5% of the variation in heart weight is explained by the variation in body weights' This restates the correlation as a proportion of explained variation, but uses 76.5% which would correspond to r, not r^2. The correct proportion explained is r^2 = 58.5%, so this option is also incorrect for misrepresenting the explained variance.
In summary, the key idea is that the correlation r = 0.765 yields an explained-variation fraction of r^2 ≈ 0.585, i.e., about 58.5%. None of the options correctly state r^2; only option a gets close but still misstates the exact figure by not using the squared value. Therefore, all options contain an incorrect interpretation of r or r^2, but option a is the closest to the correct concept if one interprets r^2 as 58.5% and omits exact accuracy in the phrasing.

The correlation coefficient between heart weight and body weight in a group of mice isr = 0.765. Using body weight as the Independent Variable we can conclude that:单项选择题

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