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.
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.)
1 answer