Questions: For a set of data, r is -0.8. One of the following statements is incorrect. Which is it? a) A change of one standard deviation in x corresponds to a decrease of 0.8 standard deviations in y. b) Sixty-four percent of the variability in y is explained by the regression of y on x. c) Sixty-four percent of the variability in x is explained by the regression of x on y. d) Eighty percent of the variability in y is explained by the regression of y on x.

Solution Steps

We are given a correlation coefficient \( r = -0.8 \) and need to identify which of the provided statements is incorrect. The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables.

Let's analyze each statement one by one:

"A change of one standard deviation in \( x \) corresponds to a decrease of 0.8 standard deviations in \( y \)."

This statement is correct because the correlation coefficient \( r = -0.8 \) indicates that for every one standard deviation increase in \( x \), \( y \) decreases by 0.8 standard deviations.

"Sixty-four percent of the variability in \( y \) is explained by the regression of \( y \) on \( x \)."

The coefficient of determination \( r^2 \) represents the proportion of the variance in the dependent variable that is predictable from the independent variable. Here, \( r^2 = (-0.8)^2 = 0.64 \), or 64%. This statement is correct.

"Sixty-four percent of the variability in \( x \) is explained by the regression of \( x \) on \( y \)."

The coefficient of determination \( r^2 \) is symmetric, meaning it applies to both \( y \) on \( x \) and \( x \) on \( y \). Thus, this statement is also correct.

"Eighty percent of the variability in \( y \) is explained by the regression of \( y \) on \( x \)."

This statement is incorrect because, as calculated earlier, \( r^2 = 0.64 \), which means 64% of the variability in \( y \) is explained by the regression of \( y \) on \( x \), not 80%.

Final Answer