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 =
To compute the correlation coefficient using the given formulas, we need to calculate several values first.
Let's start by calculating the required values:
1. Calculate the sum of each column: Σx (sum of x values) and Σy (sum of y values).
- Σx = 1 + 1 + 0 + 0 + 0 = 2
- Σy = 7 + 3 + 5 + 6 + 6 = 27
2. Calculate the sum of the product of x and y values: Σ(xy).
- Σ(xy) = (1 * 7) + (1 * 3) + (0 * 5) + (0 * 6) + (0 * 6) = 10
3. Calculate the sum of the squares of x values and y values: Σ(x^2) and Σ(y^2).
- Σ(x^2) = (1^2) + (1^2) + (0^2) + (0^2) + (0^2) = 2
- Σ(y^2) = (7^2) + (3^2) + (5^2) + (6^2) + (6^2) = 179
4. Calculate the number of data points (n).
- n = number of values in either x or y column = 5
Now, we can substitute these values into the given formula for the correlation coefficient:
r = (n * Σ(xy) - Σx * Σy) / sqrt((n * Σ(x^2) - Σx^2) * (n * Σ(y^2) - Σy^2))
Substituting the calculated values:
r = (5 * 10 - 2 * 27) / sqrt((5 * 2 - 2^2) * (5 * 179 - 27^2))
Simplifying further:
r = (50 - 54) / sqrt((10 - 4) * (895 - 729))
= -4 / sqrt(6 * 166)
= -4 / sqrt(996)
= -4 / 31.62 (approximating the square root)
Finally, calculating the correlation coefficient:
r = -0.13 (rounded to two decimal places)
Therefore, the correlation coefficient for the given data is approximately -0.13.