Consider the following sample data:
x 9 11 8 6 3
y 12 2 8 4 9
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:
1. Calculate the mean of x and y:
mean_x = (9 + 11 + 8 + 6 + 3) / 5 = 7.4
mean_y = (12 + 2 + 8 + 4 + 9) / 5 = 7
2. Subtract the mean from each value in x and y:
x - mean_x: 1.6, 3.6, 0.6, -1.4, -4.4
y - mean_y: 5, -5, 1, -3, 2
3. Multiply the differences for each pair:
(1.6 * 5) + (3.6 * -5) + (0.6 * 1) + (-1.4 * -3) + (-4.4 * 2)
= 8 - 18 + 0.6 + 4.2 - 8.8
= -13
4. Divide the sum by n-1 (where n is the sample size, which is 5 in this case):
-13 / 4 = -3.25
Therefore, the covariance between the variables is -3.25.
To calculate the correlation coefficient:
1. Calculate the standard deviation of x and y:
- Calculate the squared difference for each value in x and y:
(x - mean_x)^2: 2.56, 12.96, 0.36, 1.96, 19.36
(y - mean_y)^2: 25, 25, 1, 9, 4
- Calculate the sum of the squared differences:
2.56 + 12.96 + 0.36 + 1.96 + 19.36 = 37.2
25 + 25 + 1 + 9 + 4 = 64
- Divide the sums by n-1:
37.2 / 4 = 9.3
64 / 4 = 16
2. Calculate the square root of the standard deviations:
sqrt(9.3) = 3.05
sqrt(16) = 4
3. Divide the covariance by the product of the standard deviations:
-3.25 / (3.05 * 4) = -0.27
Therefore, the correlation coefficient is -0.27.
To calculate the covariance between the variables, we can use the formula:
Cov(X, Y) = Σ((xᵢ - μₓ)(yᵢ - μᵧ)) / (n - 1)
where Σ represents the sum, xᵢ and yᵢ represent individual data points, μₓ represents the mean of variable X, μᵧ represents the mean of variable Y, and n represents the number of data points.
First, let's calculate the means of variables X and Y:
mean X (μₓ) = (9 + 11 + 8 + 6 + 3) / 5 = 37 / 5 = 7.4
mean Y (μᵧ) = (12 + 2 + 8 + 4 + 9) / 5 = 35 / 5 = 7
Next, we calculate the deviations for each data point:
Deviation of X (xᵢ - μₓ):
9 - 7.4 = 1.6
11 - 7.4 = 3.6
8 - 7.4 = 0.6
6 - 7.4 = -1.4
3 - 7.4 = -4.4
Deviation of Y (yᵢ - μᵧ):
12 - 7 = 5
2 - 7 = -5
8 - 7 = 1
4 - 7 = -3
9 - 7 = 2
Now, we can calculate the covariance:
Cov(X, Y) = ((1.6 * 5) + (3.6 * -5) + (0.6 * 1) + (-1.4 * -3) + (-4.4 * 2)) / (5 - 1)
= (8 - 18 + 0.6 + 4.2 + -8.8) / 4
= -13 / 4
= -3.25
Therefore, the covariance between the variables X and Y is -3.25.
To calculate the correlation coefficient, we divide the covariance by the product of the standard deviations of variables X and Y.
Standard deviation of X (σₓ):
Using the formula:
σₓ = sqrt(Σ((xᵢ - μₓ)²) / (n - 1))
σₓ = sqrt((1.6² + 3.6² + 0.6² + (-1.4)² + (-4.4)²) / 4)
= sqrt((2.56 + 12.96 + 0.36 + 1.96 + 19.36) / 4)
= sqrt(37.20 / 4)
= sqrt(9.3)
≈ 3.05
Standard deviation of Y (σᵧ):
Using the same formula:
σᵧ = sqrt((5² + (-5)² + 1² + (-3)² + 2²) / 4)
= sqrt((25 + 25 + 1 + 9 + 4) / 4)
= sqrt(64 / 4)
= sqrt(16)
= 4
Now, we can calculate the correlation coefficient:
Correlation coefficient (r) = Cov(X, Y) / (σₓ * σᵧ)
= -3.25 / (3.05 * 4)
= -3.25 / 12.2
≈ -0.266
Therefore, the correlation coefficient between the variables X and Y is approximately -0.266.