Does studying for an exam pay off? The number of hours studied, x, is compared with the exam grade received, y.
x 7 7 5 5 7
y 95 90 75 85 95
(a) Complete the preliminary calculations: SS(x), SS(y), and SS(xy).
Incorrect: Your answer is incorrect. . (SS(x))
Incorrect: Your answer is incorrect. . (SS(y))
Incorrect: Your answer is incorrect. . (SS(xy))
(b) Find r. (Give your answer correct to three decimal places.)
Can someone help me and my friend we have worked this out several times come up with the same answers but it is wrong. We would like to see where we are doing it wrong.
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.