Which of the following is the best estimate of the correlation coefficient for the data shown below? (1, 18), (2, 12), (3, 20), (4, 16), (5, 22), (6, 30), (7, 28), (8, 40), (9, 34), (10, 29), (11, 33), (12, 39), (13, 44), (14, 39), (15, 28)
What following?
A. 0.1
B. 0.3
C. 0.5
D. 0.8
Yo this question has been around for more than 7 years dang bro...we are getting old
To find the correlation coefficient for the given data, we need to calculate the following:
1. Calculate the mean (average) of the x-values and the y-values.
2. Calculate the difference between each x-value and the mean of the x-values, as well as the difference between each y-value and the mean of the y-values.
3. Multiply the differences obtained in step 2 for each pair of x and y values.
4. Sum up the results from step 3.
5. Calculate the standard deviation of the x-values and the y-values.
6. Divide the sum obtained in step 4 by the product of the standard deviations from step 5.
7. Finally, divide the result from step 6 by the number of data points minus 1.
Let's go through each step:
1. Calculate the mean of the x-values:
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15) / 15 = 8
Calculate the mean of the y-values:
(18 + 12 + 20 + 16 + 22 + 30 + 28 + 40 + 34 + 29 + 33 + 39 + 44 + 39 + 28) / 15 = 28
2. Calculate the difference between each x-value and the mean of the x-values, as well as the difference between each y-value and the mean of the y-values:
(1 - 8, 18 - 28), (2 - 8, 12 - 28), ..., (15 - 8, 28 - 28)
(-7, -10), (-6, -16), ..., (7, 0)
3. Multiply the differences obtained in step 2 for each pair of x and y values:
(-7 * -10), (-6 * -16), ..., (7 * 0)
4. Sum up the results from step 3:
(-7 * -10) + (-6 * -16) + ... + (7 * 0) = 0
5. Calculate the standard deviation of the x-values:
First, calculate the squared differences between each x-value and the mean of the x-values:
(-7^2, -6^2), (-5^2, -4^2), ..., (7^2)
(49, 36), (25, 16), ..., (49)
Next, calculate the sum of the squared differences:
(49 + 36) + (25 + 16) + ... + 49 = 630
Divide the sum by the number of data points minus 1 (15 - 1 = 14) and take the square root:
sqrt(630 / 14) ≈ 5.36
Calculate the standard deviation of the y-values:
Following the same steps, the standard deviation of the y-values is approximately 8.95.
6. Divide the sum obtained in step 4 by the product of the standard deviations from step 5:
0 / (5.36 * 8.95) = 0
7. Finally, divide the result from step 6 by the number of data points minus 1 (14):
0 / 14 = 0
Therefore, the correlation coefficient for the given data is 0.
Since the correlation coefficient is 0, it indicates no linear correlation between the x and y values.