To complete the preliminary calculations, we need to calculate the sum of squares for x (SS(x)), y (SS(y)), and xy (SS(xy)).
Here's how to do it step by step:
(a) Calculating SS(x):
SS(x) = Σ(x - x̄)²
First, we need to calculate the mean (x̄) of the values in x:
x = 7, 7, 5, 5, 7
x̄ = (7 + 7 + 5 + 5 + 7) / 5 = 31 / 5 = 6.2
Now, calculate the squared deviations for each value in x and sum them up:
(x₁ - x̄)² = (7 - 6.2)² = 0.64
(x₂ - x̄)² = (7 - 6.2)² = 0.64
(x₃ - x̄)² = (5 - 6.2)² = 1.44
(x₄ - x̄)² = (5 - 6.2)² = 1.44
(x₅ - x̄)² = (7 - 6.2)² = 0.64
Summing up the squared deviations:
SS(x) = 0.64 + 0.64 + 1.44 + 1.44 + 0.64 = 4
Now let's move on to calculating SS(y) and SS(xy):
(b) Calculating SS(y):
SS(y) = Σ(y - ȳ)²
First, calculate the mean (ȳ) of the values in y:
y = 95, 90, 75, 85, 95
ȳ = (95 + 90 + 75 + 85 + 95) / 5 = 440 / 5 = 88
Next, calculate the squared deviations for each value in y and sum them up:
(y₁ - ȳ)² = (95 - 88)² = 49
(y₂ - ȳ)² = (90 - 88)² = 4
(y₃ - ȳ)² = (75 - 88)² = 169
(y₄ - ȳ)² = (85 - 88)² = 9
(y₅ - ȳ)² = (95 - 88)² = 49
Summing up the squared deviations:
SS(y) = 49 + 4 + 169 + 9 + 49 = 280
(c) Calculating SS(xy):
SS(xy) = Σ[(x - x̄)(y - ȳ)]
Calculate the deviations for each pair of (x, y) values and sum the products:
(x₁ - x̄)(y₁ - ȳ) = (7 - 6.2)(95 - 88) = 6 * 7 = 42
(x₂ - x̄)(y₂ - ȳ) = (7 - 6.2)(90 - 88) = 6 * 2 = 12
(x₃ - x̄)(y₃ - ȳ) = (5 - 6.2)(75 - 88) = -2.2 * -13 = 28.6
(x₄ - x̄)(y₄ - ȳ) = (5 - 6.2)(85 - 88) = -1.2 * -3 = 3.6
(x₅ - x̄)(y₅ - ȳ) = (7 - 6.2)(95 - 88) = 6 * 7 = 42
Summing up the products:
SS(xy) = 42 + 12 + 28.6 + 3.6 + 42 = 128.2
Now, let's move on to finding r (the correlation coefficient).
(b) Finding r:
r = SS(xy) / √(SS(x) * SS(y))
Using the previously calculated values:
r = 128.2 / √(4 * 280) ≈ 0.591
Therefore, the correlation coefficient (r) is approximately 0.591.
Make sure to double-check your calculations to ensure accuracy. If you are still having difficulties, please let me know, and I'll be happy to assist you further.