A sample of eight observations of variables x and y is shown below:
x 5 3 7 9 2 4 6 8
y 20 23 15 11 27 21 17 14
Find the value of coefficient of correlation, r.
a) - 0.991
b)0.872
c)- 0.512
d)0.942
I'll narrow it down for you. It's either a) or d).
If you need to do this by hand, find a correlation coefficient formula, plug in the values and go from there to find r.
To calculate the coefficient of correlation (r), you need to follow these steps:
Step 1: Calculate the mean of x and y.
- Sum all the values of x: 5 + 3 + 7 + 9 + 2 + 4 + 6 + 8 = 44
- Divide the sum by the number of observations (n), in this case, 8: 44 / 8 = 5.5
- The mean of x is 5.5.
- Sum all the values of y: 20 + 23 + 15 + 11 + 27 + 21 + 17 + 14 = 128
- Divide the sum by 8: 128 / 8 = 16
- The mean of y is 16.
Step 2: Calculate the product of each observation with the mean.
- Subtract the mean of x from each x value and the mean of y from each y value.
- Multiply each of these differences together.
- For example, for the first pair (5, 20): (5 - 5.5) * (20 - 16) = (-0.5) * (4) = -2
Step 3: Calculate the sum of all the products obtained in Step 2.
- Sum all the products calculated in Step 2.
- For the given data set, the sum of all products = -2 + 0.5 + 5.5 - 6 + 3 + 1.5 - 0.5 - 1 = 1.
Step 4: Calculate the square of each x value and each y value.
- For example, the square of 5 is 5^2 = 25, and the square of 20 is 20^2 = 400.
Step 5: Calculate the sum of the squares for both x and y.
- Sum all the squared values of x: 25 + 9 + 49 + 81 + 4 + 16 + 36 + 64 = 284
- Sum all the squared values of y: 400 + 529 + 225 + 121 + 729 + 441 + 289 + 196 = 2930
Step 6: Calculate the square root of the product of the sums of the squares from Step 5.
- Take the square root of (sum of squared x) * (sum of squared y): sqrt(284 * 2930) ≈ 634.474
Step 7: Calculate the square root of the product of the sums of the squares from Step 5.
- Multiply the sum of the products from Step 3 by 8 (number of observations): 1 * 8 = 8.
Step 8: Divide the result from Step 7 by the result from Step 6 to find the coefficient of correlation, r.
- Divide 8 by 634.474: 8 / 634.474 ≈ 0.012
After rounding to three decimal places, the coefficient of correlation (r) is approximately 0.012.
Therefore, none of the options a), b), c), or d) provided fit the value calculated.