Question

Suppose a correlation is computed in each of two samples. If the value of SSXY is the same in each sample, and the square root of SSXSSY is larger in sample 1, then in which sample will the value of the correlation coefficient be larger?

a. Sample 1.
b. Sample 2.
c. None; the correlation coefficient will be the same in both samples.
d. There is not enough information to answer this question.

Answer 1

The correlation coefficient will be larger in Sample 2 because the correlation coefficient is inversely proportional to the square root of SSXSSY, and this value is larger in Sample 1.

Step-by-step explanation:

If the value of SSXY is the same in each sample and the square root of SSXSSY is larger in sample 1, then the correlation coefficient, which is calculated using SSXY divided by the square root of SSXSSY, will be larger in the sample with the smaller denominator, assuming the numerators are equal. Hence, the correlation coefficient will be larger in sample 2 because the denominators in sample 1 are larger and the correlation coefficient is inversely proportional to the denominator given a constant numerator.

To understand this, we must recall how the correlation coefficient (r) is calculated. This coefficient indicates the strength and direction of a linear relationship between two variables. A key equation for calculating this is: r = SSXY / √(SSX * SSY). Here, if SSXY is constant and √(SSX * SSY) is larger in one sample compared to another, the correlation coefficient, r, will be smaller in the sample with the larger square root of SSXSSY.

Thus, the correct answer to the question is b. Sample 2. The correlation coefficient for sample 2 must be larger since the square root of SSXSSY, the divisor in the correlation calculation, is smaller than that in sample 1.