use the table of values to calculate the linear correlation coefficient r
x -20 3 -7 10 6
y -8 1 -9 6 -5
a) r = 0.968
b) r = 0.781
c) r = 0.625
d) r = 0.756
To calculate the linear correlation coefficient (r) using the given table of values, follow these steps:
Step 1: Calculate the mean of x (x̄) and the mean of y (ȳ):
x̄ = (-20 + 3 - 7 + 10 + 6) / 5 = -8 / 5 = -1.6
ȳ = (-8 + 1 - 9 + 6 - 5) / 5 = -15 / 5 = -3
Step 2: Calculate the deviations from the mean for each x and y value.
For x:
x - x̄: -20 - (-1.6) = -20 + 1.6 = -18.4
3 - (-1.6) = 3 + 1.6 = 4.6
-7 - (-1.6) = -7 + 1.6 = -5.4
10 - (-1.6) = 10 + 1.6 = 11.6
6 - (-1.6) = 6 + 1.6 = 7.6
For y:
y - ȳ: -8 - (-3) = -8 + 3 = -5
1 - (-3) = 1 + 3 = 4
-9 - (-3) = -9 + 3 = -6
6 - (-3) = 6 + 3 = 9
-5 - (-3) = -5 + 3 = -2
Step 3: Square the deviations for both x and y.
(x - x̄)^2: (-18.4)^2 ≈ 338.56
(4.6)^2 ≈ 21.16
(-5.4)^2 ≈ 29.16
(11.6)^2 ≈ 134.56
(7.6)^2 ≈ 57.76
(y - ȳ)^2: (-5)^2 = 25
4^2 = 16
(-6)^2 = 36
9^2 = 81
(-2)^2 = 4
Step 4: Calculate the sum of the squared deviations for both x and y.
Sum of (x - x̄)^2 = 338.56 + 21.16 + 29.16 + 134.56 + 57.76 ≈ 581.2
Sum of (y - ȳ)^2 = 25 + 16 + 36 + 81 + 4 = 162
Step 5: Calculate the square root of the product of the sums of squared deviations.
√(Sum of (x - x̄)^2 * Sum of (y - ȳ)^2) = √(581.2 * 162) ≈ √94195.2 ≈ 307
Step 6: Calculate the sum of the product of the deviations of x and y.
(x - x̄)(y - ȳ): (-18.4 * -5) + (4.6 * 4) + (-5.4 * -6) + (11.6 * 9) + (7.6 * -2) = 92 + 18.4 - 32.4 + 104.4 - 15.2 ≈ 167.2
Step 7: Calculate the linear correlation coefficient (r).
r = (sum of the product of the deviations of x and y) / (square root of the product of the sums of squared deviations)
r = 167.2 / 307 ≈ 0.544
Therefore, the linear correlation coefficient (r) for the given table of values is approximately 0.544.
The correct answer is not among the answer choices and should be presented as: r ≈ 0.544
To calculate the linear correlation coefficient (r) using the table of values, you can follow these steps:
Step 1: Calculate the mean (average) of both x and y values.
The mean for x can be calculated as:
Mean (x) = (x1 + x2 + x3 + x4 + x5) / 5 = (-20 + 3 - 7 + 10 + 6) / 5 = -8 / 5 = -1.6
Similarly, the mean for y can be calculated as:
Mean (y) = (y1 + y2 + y3 + y4 + y5) / 5 = (-8 + 1 - 9 + 6 - 5) / 5 = -15 / 5 = -3
Step 2: Calculate the differences between each x value and the mean of x (x - mean(x)), as well as the differences between each y value and the mean of y (y - mean(y)).
For x: -20 - (-1.6) = -20 + 1.6 = -18.4
For y: -8 - (-3) = -8 + 3 = -5
For x: 3 - (-1.6) = 3 + 1.6 = 4.6
For y: 1 - (-3) = 1 + 3 = 4
For x: -7 - (-1.6) = -7 + 1.6 = -5.4
For y: -9 - (-3) = -9 + 3 = -6
For x: 10 - (-1.6) = 10 + 1.6 = 11.6
For y: 6 - (-3) = 6 + 3 = 9
For x: 6 - (-1.6) = 6 + 1.6 = 7.6
For y: -5 - (-3) = -5 + 3 = -2
Step 3: Calculate the product of the differences for each pair of x and y values (x - mean(x)) * (y - mean(y)).
For the first pair: (-18.4) * (-5) = 92
For the second pair: (4.6) * (4) = 18.4
For the third pair: (-5.4) * (-6) = 32.4
For the fourth pair: (11.6) * (9) = 104.4
For the fifth pair: (7.6) * (-2) = -15.2
Step 4: Calculate the sum of all the products obtained in Step 3.
Sum of products = 92 + 18.4 + 32.4 + 104.4 - 15.2 = 232
Step 5: Calculate the squares of the differences for each x value and the mean of x, as well as the squares of the differences for each y value and the mean of y [(x - mean(x))^2 and (y - mean(y))^2].
For x: (-18.4)^2 = 338.56
For y: (-5)^2 = 25
For x: (4.6)^2 = 21.16
For y: (4)^2 = 16
For x: (-5.4)^2 = 29.16
For y: (-6)^2 = 36
For x: (11.6)^2 = 134.56
For y: (9)^2 = 81
For x: (7.6)^2 = 57.76
For y: (-2)^2 = 4
Step 6: Calculate the sum of all the squares obtained in Step 5.
Sum of x squares = 338.56 + 21.16 + 29.16 + 134.56 + 57.76 = 581.2
Sum of y squares = 25 + 16 + 36 + 81 + 4 = 162
Step 7: Calculate the product of the square root of the sum of x squares and the square root of the sum of y squares, sqrt(Sum of x squares) * sqrt(Sum of y squares).
sqrt(581.2) * sqrt(162) = 24.109 * 12.727 = 307.71
Step 8: Calculate the square root of the product obtained in Step 7, sqrt(Sum of x squares) * sqrt(Sum of y squares).
r = 232 / 307.71 = 0.754
Therefore, the linear correlation coefficient (r) is approximately 0.754.
None of the provided options match the calculated r value.