Understanding Pearson's Correlation and Linear Transformations

The question asks how the Pearson's correlation coefficient changes when the observations of the variables X and Y are transformed linearly. We are given the original correlation coefficient between X and Y is -0.8.

Effect of Linear Transformations on Pearson's Correlation

Pearson's correlation coefficient measures the strength and direction of a linear relationship between two variables. A key property of this coefficient is how it behaves under linear transformations.

Let's consider two variables X and Y with Pearson's correlation coefficient \(r_{XY}\). Suppose we transform these variables linearly to get new variables X' and Y':

where a, b, c, and d are constants. The Pearson's correlation coefficient between the new variables X' and Y', denoted as \(r_{X'Y'}\), is related to the original correlation coefficient by the formula:

The term \(\frac{ac}{|ac|}\) is equal to 1 if the product of the scaling factors \(ac\) is positive, and -1 if \(ac\) is negative. This means:

• If \(a\) and \(c\) have the same sign (both positive or both negative), then \(ac > 0\), so \(\frac{ac}{|ac|} = 1\). In this case, \(r_{X'Y'} = r_{XY}\). The correlation coefficient remains the same.
• If \(a\) and \(c\) have different signs (one positive and one negative), then \(ac < 0\), so \(\frac{ac}{|ac|} = -1\). In this case, \(r_{X'Y'} = -r_{XY}\). The sign of the correlation coefficient flips.

The constant terms \(b\) and \(d\) (shifts in origin) do not affect the correlation coefficient.

Applying the Transformation to the Problem

In this specific problem, the original variables are X and Y, and their correlation is given as \(r_{XY} = -0.8\).

The new variables X' and Y' are obtained by:

• Each observation of X is halved. This means the transformation for X is \(X' = \frac{1}{2}X\). Comparing this to \(X' = aX + b\), we have \(a = \frac{1}{2}\) and \(b = 0\). The scaling factor for X is \(a = \frac{1}{2}\), which is positive.
• Each observation of Y is doubled. This means the transformation for Y is \(Y' = 2Y\). Comparing this to \(Y' = cY + d\), we have \(c = 2\) and \(d = 0\). The scaling factor for Y is \(c = 2\), which is positive.

Now, let's find the product of the scaling factors \(a\) and \(c\):

Since \(ac = 1\), which is positive (\(ac > 0\)), the factor \(\frac{ac}{|ac|}\) is:

Using the formula \(r_{X'Y'} = \frac{ac}{|ac|} r_{XY}\), we get:

Since the original correlation coefficient \(r_{XY}\) is given as -0.8, the new Pearson's correlation coefficient \(r_{X'Y'}\) is:

Conclusion

When each observation of X is halved (scaled by a positive factor 1/2) and each observation of Y is doubled (scaled by a positive factor 2), the Pearson's correlation coefficient remains unchanged because both scaling factors are positive. The new correlation coefficient is therefore the same as the original one, which is -0.8.

The calculated new Pearson's correlation coefficient is -0.80, which corresponds to one of the given options.

Revision Table: Pearson's Correlation Transformation

Additional Information on Correlation Coefficient

The Pearson's correlation coefficient, often denoted by \(r\), is a standardized measure of the linear association between two continuous variables. It ranges from -1 to +1.

• A value of +1 indicates a perfect positive linear relationship.
• A value of -1 indicates a perfect negative linear relationship.
• A value of 0 indicates no linear relationship.

It is a dimensionless quantity, meaning it does not have the units of the variables being measured. This makes it useful for comparing the strength of relationships between different pairs of variables.

One of the properties that makes Pearson's r useful is its invariance under positive linear transformations. This means that if you change the scale or shift the origin of either variable (or both), the correlation coefficient will remain the same, provided the scaling factors are positive. If one or both scaling factors are negative, the sign of the correlation changes, reflecting a reversal in the direction of the relationship.