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.)
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.
To calculate the covariance between the variables, follow these steps:
a. Calculate the mean of each variable:
x: (25 + 18 + 20 + 12 + 22) / 5 = 19.40
y: (27 + 20 + 25 + 15 + 22) / 5 = 21.80
b. Calculate the deviations from the mean for each data point:
For x: (25 - 19.40), (18 - 19.40), (20 - 19.40), (12 - 19.40), (22 - 19.40)
= 5.60, -1.40, 0.60, -7.40, 2.60
For y: (27 - 21.80), (20 - 21.80), (25 - 21.80), (15 - 21.80), (22 - 21.80)
= 5.20, -1.80, 3.20, -6.80, 0.20
c. Multiply the deviations for each data point:
(5.60 * 5.20), (-1.40 * -1.80), (0.60 * 3.20), (-7.40 * -6.80), (2.60 * 0.20)
= 29.12, 2.52, 1.92, 50.32, 0.52
d. Calculate the average of the multiplied deviations:
(29.12 + 2.52 + 1.92 + 50.32 + 0.52) / 5
= 16.28
Therefore, the covariance between the variables is 16.28.
Now, to calculate the correlation coefficient,
e. Calculate the standard deviation for each variable:
Standard deviation of x = sqrt((29.12 + 2.52 + 1.92 + 50.32 + 0.52) / (5-1))
= sqrt(84.40 / 4) = sqrt(21.10) = 4.59
Standard deviation of y = sqrt((5.20^2 + (-1.80)^2 + 3.20^2 + (-6.80)^2 + 0.20^2) / (5-1))
= sqrt(84.40 / 4) = sqrt(21.10) = 4.59
f. Calculate the product of the standard deviations:
4.59 * 4.59 = 21.11
g. Calculate the correlation coefficient:
Correlation coefficient = Covariance / (Standard deviation of x * Standard deviation of y)
= 16.28 / 21.11 = 0.7726
Therefore, the correlation coefficient between the variables is 0.77.