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