Question 1187210: Q No.6)
Part a)
The following table shows verbal reasoning test score, x, and an English test score, y, for each of a random sample of 8 children who took both tests.
Child A B C D E F G H
x 112 113 110 113 112 114 109 113
y 69 65 75 70 70 75 68 76
Calculate the Co-efficient of correlation “r” between the scores in verbal reasoning and English.
Part b)
The age (X) and systolic blood pressure (Y) of 100 individuals gave the following sums:
X i 4421, Yi 12130, X i2 208349, Yi2 1498976, X iYi 542735
Determine the equation of the least squares regression line of Y on X.
the missing letter is sumation (E)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Part a) Correlation Coefficient (r)
Here's how to calculate the correlation coefficient (r) for the verbal reasoning (x) and English (y) test scores:
Calculate the means of x and y:
Mean of x (x̄) = (112 + 113 + 110 + 113 + 112 + 114 + 109 + 113) / 8 = 111.75
Mean of y (ȳ) = (69 + 65 + 75 + 70 + 70 + 75 + 68 + 76) / 8 = 71
Calculate the deviations from the mean for x and y:
For each x value, calculate (x - x̄)
For each y value, calculate (y - ȳ)
Child x y x - x̄ y - ȳ
A 112 69 0.25 -2
B 113 65 1.25 -6
C 110 75 -1.75 4
D 113 70 1.25 -1
E 112 70 0.25 -1
F 114 75 2.25 4
G 109 68 -2.75 -3
H 113 76 1.25 5
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Calculate the products of the deviations:
Multiply (x - x̄) and (y - ȳ) for each child.
Child x - x̄ y - ȳ (x - x̄)(y - ȳ)
A 0.25 -2 -0.5
B 1.25 -6 -7.5
C -1.75 4 -7
D 1.25 -1 -1.25
E 0.25 -1 -0.25
F 2.25 4 9
G -2.75 -3 8.25
H 1.25 5 6.25
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Calculate the sum of the products of deviations (Σ(x - x̄)(y - ȳ)):
Sum the values in the last column: -0.5 + (-7.5) + (-7) + (-1.25) + (-0.25) + 9 + 8.25 + 6.25 = 7
Calculate the sum of squared deviations for x and y:
Σ(x - x̄)² = (0.25)² + (1.25)² + (-1.75)² + (1.25)² + (0.25)² + (2.25)² + (-2.75)² + (1.25)² = 14.75
Σ(y - ȳ)² = (-2)² + (-6)² + 4² + (-1)² + (-1)² + 4² + (-3)² + 5² = 96
Calculate the correlation coefficient (r):
r = Σ(x - x̄)(y - ȳ) / sqrt[Σ(x - x̄)² * Σ(y - ȳ)²]
r = 7 / sqrt(14.75 * 96)
r = 7 / sqrt(1416)
r ≈ 7 / 37.63
r ≈ 0.186
Part b) Least Squares Regression Line
The equation of the least squares regression line of Y on X is given by:
y = a + bx
where:
b = (ΣXᵢYᵢ - (ΣXᵢ * ΣYᵢ) / n) / (ΣXᵢ² - (ΣXᵢ)² / n)
a = ȳ - b * x̄
Given:
ΣXᵢ = 4421
ΣYᵢ = 12130
ΣXᵢ² = 208349
ΣYᵢ² = 1498976
ΣXᵢYᵢ = 542735
n = 100
Calculate b:
b = (542735 - (4421 * 12130) / 100) / (208349 - (4421)² / 100)
b = (542735 - 536497.3) / (208349 - 195452.41)
b = 6237.7 / 12896.59
b ≈ 0.484
Calculate the means of X and Y:
x̄ = ΣXᵢ / n = 4421 / 100 = 44.21
ȳ = ΣYᵢ / n = 12130 / 100 = 121.3
Calculate a:
a = ȳ - b * x̄
a = 121.3 - (0.484 * 44.21)
a = 121.3 - 21.41
a ≈ 99.89
Therefore, the equation of the least squares regression line of Y on X is:
y = 99.89 + 0.484x
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