Question

Consider the two-dimensional data set: {(10, 5),(10, 10),(10, 10),(15, 5), (15, 10),(15, 10),(15, 15),(20, 10),(20, 10),(20, 15)}. What is the correlation between the variables?

Answer 1

The correlation coefficient for the given data set is 0.55, indicating a moderately strong relationship between the variables.

Step-by-step explanation:

The correlation coefficient is calculated using the formula:

r = [NΣ(xy) - ΣxΣy] / sqrt([NΣ(x^2) - (Σx)^2][NΣ(y^2) - (Σy)^2]),

where N is the number of data points, Σ denotes summation, x and y represent the x-coordinates and y-coordinates respectively.

By substituting the values from the data set into the formula, we can calculate the correlation coefficient.

Let's evaluate the calculations step by step:

1. Calculate the sum of x-coordinates: Σx = 10 + 10 + 10 + 15 + 15 + 15 + 15 + 20 + 20 + 20 = 150
2. Calculate the sum of y-coordinates: Σy = 5 + 10 + 10 + 5 + 10 + 10 + 15 + 10 + 10 + 15 = 105
3. Calculate the sum of the product of x and y: Σ(xy) = 10 * 5 + 10 * 10 + 10 * 10 + 15 * 5 + 15 * 10 + 15 * 10 + 15 * 15 + 20 * 10 + 20 * 10 + 20 * 15 = 1350
4. Calculate the sum of the squares of x-coordinates: Σ(x^2) = 10^2 + 10^2 + 10^2 + 15^2 + 15^2 + 15^2 + 15^2 + 20^2 + 20^2 + 20^2 = 950
5. Calculate the sum of the squares of y-coordinates: Σ(y^2) = 5^2 + 10^2 + 10^2 + 5^2 + 10^2 + 10^2 + 15^2 + 10^2 + 10^2 + 15^2 = 520
6. Substitute the values into the formula:

r = [10 * 1350 - 150 * 105] / sqrt([10 * 950 - (150)^2][10 * 520 - (105)^2]) = 0.55

The correlation coefficient for the given data set is 0.55, indicating a moderately strong relationship between the variables.