Calculation of Correlation Coefficient

The correlation coefficient (r) between two variables x and y can be calculated using the following formula:

r = (n∑XY – ∑X∑Y) / [√(n∑X^2 – (∑X)^2) √(n∑Y^2 – (∑Y)^2)]

where,
n = sample size
∑XY = sum of the products of corresponding values of x and y
∑X = sum of values of x
∑Y = sum of values of y
∑X^2 = sum of squares of values of x
∑Y^2 = sum of squares of values of y

Given that,
Average of x (x̄) = 8 – 10.5 = -2.5
Average of y (ȳ) = 13.9
Standard deviation of x (σx) = 3.5
Standard deviation of y (σy) = 4.1
n = 10

We need to find the value of ∑XY. We can use the following formula to calculate it:

∑XY = (n∑x y – ∑x ∑y) / n
∑XY = (10 * ȳ * x̄ – 10 * 8 * 13.9) / 10
∑XY = -347

Now, we can substitute the given values in the formula of correlation coefficient to get:

r = (10*-347 - 8*13.9*-2.5) / [√(10*(8^2) – (8)^2) √(10*(13.9^2) – (13.9)^2)]
r = -3665 / [√(640) √(1764)]
r = -3665 / [25.298 42]
r = -0.656

Therefore, the correlation coefficient between x and y is -0.656, which is closest to -0.66. Hence, option (b) is the correct answer.

Interpretation of Correlation Coefficient

The correlation coefficient (r) is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation.

In this case, the correlation coefficient (-0.656) is negative, which indicates a negative correlation between x and y. It means that as the values of x increase, the values of y tend to decrease, and vice versa. The magnitude of -0.656 suggests a moderate negative correlation between x and y.

However, it is important to note that correlation does not imply causation. A negative correlation between x and y does not necessarily mean that x causes y to decrease or vice versa. There could be other factors that influence the relationship between x and y, which need to be investigated further.