The Correct Answer and Explanation is:
The correct answer is 0.88.
The solution to this problem lies in understanding a core property of the Pearson correlation coefficient (r). The correlation coefficient is a measure of the linear relationship between two variables, and its value is unaffected by linear transformations of those variables.
A linear transformation is any change to a variable that involves adding, subtracting, multiplying, or dividing by a constant. Converting a raw score (X) into a z-score is a classic example of a linear transformation. The formula for a z-score is z = (X – μ) / σ, where μ is the mean and σ is the standard deviation. This formula involves subtracting a constant (the mean) and dividing by another constant (the standard deviation).
The key concept is that the correlation coefficient is invariant to such changes. Think of it like changing units of measurement. The correlation between height and weight is the same whether you measure height in inches and weight in pounds, or height in centimeters and weight in kilograms. You are simply relabeling the axes, not changing the fundamental linear pattern of the data points.
In this problem, the initial correlation between scores X and Y is given as 0.88. When both X and Y are converted to z-scores, they are each being linearly transformed. This process standardizes the variables, giving them a mean of 0 and a standard deviation of 1, but it does not alter the strength or the direction of the linear relationship that exists between them. Since the transformation for both variables involves dividing by the standard deviation (a positive number), the direction of the relationship is preserved. Therefore, the correlation between the z-scores of X and the z-scores of Y will be identical to the original correlation between X and Y. The new correlation remains 0.88.
