A set of data is given as follows: X = {3, 4, 5, 6, 7} and Y = {2, 5, 7, 8, 9}. Determine if the correlation between X and Y is significant.
Solution
Step 1 ：Step 1: Calculate the sample size (n), which is the number of pairs of data. In this case, \( n = 5 \).
Step 2 ：Step 2: Calculate the sum of X (\( \Sigma X \)), the sum of Y (\( \Sigma Y \)), the sum of XY (\( \Sigma XY \)), the square sum of X (\( \Sigma X^2 \)), and the square sum of Y (\( \Sigma Y^2 \)). Here, \( \Sigma X = 25 \), \( \Sigma Y = 31 \), \( \Sigma XY = 145 \), \( \Sigma X^2 = 89 \), and \( \Sigma Y^2 = 219 \).
Step 3 ：Step 3: Substitute these values into the correlation coefficient formula \( r = \frac{n(\Sigma XY) - (\Sigma X)(\Sigma Y)}{\sqrt{[n(\Sigma X^2) - (\Sigma X)^2][n(\Sigma Y^2) - (\Sigma Y)^2]}} \). This yields \( r = \frac{5(145) - (25)(31)}{\sqrt{[5(89) - (25)^2][5(219) - (31)^2]}} \).
Step 4 ：Step 4: Simplify the above equation to find the value of r.
Step 5 ：Step 5: Compare the calculated r value with the critical value (for a two-tailed test with \( \alpha = 0.05 \) and degrees of freedom \( df = n - 2 = 3 \), the critical value is approximately 0.878). If the absolute value of r is greater than the critical value, the correlation is significant.
