Question

Consider the following sample data:


x 25 18 20 12 22
y 27 20 25 15 22


picture Click here for the Excel Data File



a. Calculate the covariance between the variables. (Negative value should be indicated by a minus sign. Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)





b. Calculate the correlation coefficient. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Answer 1

To calculate the covariance between the variables, we can use the following formula:

Covariance = SUM((x - mean(x)) * (y - mean(y))) / (n - 1)

First, we need to calculate the mean of x and y.

Mean(x) = (25 + 18 + 20 + 12 + 22) / 5 = 19.4
Mean(y) = (27 + 20 + 25 + 15 + 22) / 5 = 21.8

Next, we calculate the deviations from the mean for x and y.

Deviation from mean for x:
25 - 19.4 = 5.6
18 - 19.4 = -1.4
20 - 19.4 = 0.6
12 - 19.4 = -7.4
22 - 19.4 = 2.6

Deviation from mean for y:
27 - 21.8 = 5.2
20 - 21.8 = -1.8
25 - 21.8 = 3.2
15 - 21.8 = -6.8
22 - 21.8 = 0.2

Now, we calculate the product of the deviations for each pair of observations.

Product of deviations:
(5.6 * 5.2) + (-1.4 * -1.8) + (0.6 * 3.2) + (-7.4 * -6.8) + (2.6 * 0.2) = 31.012

Finally, we divide the sum of the products by (n-1) to get the covariance.

Covariance = 31.012 / (5 - 1) = 7.753

So, the covariance between the variables is 7.753.

To calculate the correlation coefficient, we can use the formula:

Correlation coefficient = Covariance / (std(x) * std(y))

First, we need to calculate the standard deviation of x and y.

Standard deviation of x:
sqrt(((25 - 19.4)^2 + (18 - 19.4)^2 + (20 - 19.4)^2 + (12 - 19.4)^2 + (22 - 19.4)^2) / (5 - 1)) = sqrt(166.64) = 12.917

Standard deviation of y:
sqrt(((27 - 21.8)^2 + (20 - 21.8)^2 + (25 - 21.8)^2 + (15 - 21.8)^2 + (22 - 21.8)^2) / (5 - 1)) = sqrt(33.36) = 5.772

Now, we can calculate the correlation coefficient.

Correlation coefficient = 7.753 / (12.917 * 5.772) = 0.106

Therefore, the correlation coefficient is 0.106.