Suppose the correlation between two variables (x, y) in a data set is determined to be r = 0.63, What must be true about the slope, b, of the least-squares line estimated for the same set of data? The slope b is always equal to the square of the correlation r. The slope will also be a value between −1 and 1. The slope will have the same sign as the correlation. The slope will have the opposite sign as the correlation
Question
Suppose the correlation between two variables (x, y) in a data set is determined to be r = 0.63, What must be true about the slope, b, of the least-squares line estimated for the same set of data? The slope b is always equal to the square of the correlation r. The slope will also be a value between −1 and 1. The slope will have the same sign as the correlation. The slope will have the opposite sign as the correlation
Solution
The statement that the slope, b, is always equal to the square of the correlation, r, is not correct. The square of the correlation coefficient (r^2) is the coefficient of determination, which represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.
The statement that the slope will also be a value between -1 and 1 is not necessarily true. The slope of the regression line can be any real number and is not restricted to the range of -1 to 1. It depends on the units in which x and y are measured.
The statement that the slope will have the same sign as the correlation is generally true. A positive correlation coefficient indicates that as the value of x increases, the value of y also increases, which means the slope of the line is positive. Conversely, a negative correlation coefficient indicates that as the value of x increases, the value of y decreases, which means the slope of the line is negative.
The statement that the slope will have the opposite sign as the correlation is not correct. As mentioned above, the slope of the regression line generally has the same sign as the correlation coefficient.
Similar Questions
What can we say about the relationship between the correlation r and the slope b of the least-squares line for the same set of data? Both r and b always have values between −1 and 1. b is always larger than r. r is always larger than b. r and b have the same sign (+ or −). The slope b is always equal to the square of the correlation r.
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