Question 1187210
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