Consider the following table and use the formulas that are given for computing the correlation coefficient. (Give your answer correct to two decimal places.)
x 1 1 0 0 0
y 7 3 5 6 6
r =
n
xy
−
x
y
n
x2
−
x
2
n
y2
−
y
2
(1)
r =
SS(xy)
SS(x) · SS(y)
(2) r = 1.00 Incorrect: Your answer is incorrect. .
The answer should be a negative number as the numbers increase as x becomes smaller.
The correlation coefficient lies between -0.2 and -0.3.
Please post your equation for r again to see if it is interpreted correctly.
Most of the time mistakes come from incorrect interpretation of expressions.
To compute the correlation coefficient using the given formulas, we need to calculate several values first:
1. Calculate the sum of x and y:
- sum of x (Σx) = 1 + 1 + 0 + 0 + 0 = 2
- sum of y (Σy) = 7 + 3 + 5 + 6 + 6 = 27
2. Calculate the sum of the product of x and y:
- sum of xy (Σxy) = (1 * 7) + (1 * 3) + (0 * 5) + (0 * 6) + (0 * 6) = 10
3. Calculate the sum of the squares of x and y:
- sum of x squared (Σx^2) = (1^2) + (1^2) + (0^2) + (0^2) + (0^2) = 2
- sum of y squared (Σy^2) = (7^2) + (3^2) + (5^2) + (6^2) + (6^2) = 139
4. Calculate the number of observations (n):
- n = number of data points = 5
Now we can use the formulas to calculate the correlation coefficient:
1. Using Formula (1):
- r = (n*Σxy - Σx*Σy) / sqrt((n*Σx^2 - (Σx)^2) * (n*Σy^2 - (Σy)^2))
- r = (5*10 - 2*27) / sqrt((5*2 - (2)^2) * (5*139 - (27)^2))
- r = (50 - 54) / sqrt((10 - 4) * (695 - 729))
- r = -4 / sqrt(6 * (-34))
- r = -4 / sqrt(-204)
- r = -4 / -14.28
- r ≈ 0.28 (correct to two decimal places)
2. Using Formula (2):
- r = sqrt(SS(xy) / (SS(x) * SS(y)))
- r = sqrt(10 / (2 * 139))
- r = sqrt(10 / 278)
- r ≈ 1.00 (correct to two decimal places)
Therefore, the correlation coefficient, r, is approximately 0.28 (using Formula 1) or 1.00 (using Formula 2).