The Correct Answer and Explanation is:
Correct Answer: 0.88
The correct answer is 0.88. The reason for this lies in a fundamental property of the Pearson correlation coefficient: it is invariant under linear transformations of the variables.
Converting a set of raw scores into z-scores is a specific type of linear transformation. The formula for a z-score is z = (x – μ) / σ, where ‘x’ is the raw score, ‘μ’ is the mean of the scores, and ‘σ’ is the standard deviation. In this process, each score is first shifted by subtracting a constant (the mean) and then scaled by dividing by another constant (the standard deviation).
This transformation standardizes the data, giving each variable a new mean of 0 and a new standard deviation of 1. However, it does not alter the underlying linear relationship between the two variables. The correlation coefficient is a measure of the strength and direction of this linear relationship. Since the z-score transformation only changes the center and scale of the data without distorting the pattern of the data points relative to each other, the correlation remains unchanged.
Imagine a scatter plot of the original scores, X and Y. The points would form a pattern that reflects their 0.88 correlation. If you then create a new scatter plot using the z-scores of X and Y, the pattern of the points would look identical. The only difference would be the numbers on the axes, which would now be centered around zero. Because the shape and tightness of the cluster of points do not change, the measure of their linear association, the correlation coefficient, must also remain the same. Therefore, the correlation between the z-scores for X and the z-scores for Y is still 0.88.
