In the table below, the gross values of milk produced (in millions of Rand) and cattle raised (in millions) in South Africa, for ten years, are given. The relationship between milk produced (x) and cattle raised (y) is investigated.

Question

X
28.7 36.3 41.3 41.1 45.5 47.7 54.7 56.9 71.2 71.8
Y
17.7 21.4 21.2 23.7 23.8 22.3 22.3 22.9 25.2 27.1


The following information is known:

∑ x = 495.2 ∑ y = 227.6 ∑ x2 = 26336.2
∑ y2 = 5237.26 ∑ xy = 11543.89

Answer 1

To investigate the relationship between milk produced (x) and cattle raised (y), we can use the formula for calculating the correlation coefficient.

The formula is:

r = (nΣxy - ΣxΣy) / sqrt[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]

where
n = number of data points
Σ = sum of
x = milk produced
y = cattle raised

Given the values:

n = 10
Σx = 495.2
Σy = 227.6
Σx^2 = 26336.2
Σy^2 = 5237.26
Σxy = 11543.89

We can substitute these values into the formula to calculate the correlation coefficient:

r = (10 * 11543.89 - 495.2 * 227.6) / sqrt[(10 * 26336.2 - (495.2)^2)(10 * 5237.26 - (227.6)^2)]

Calculated values:
r = (115438.9 - 112767.52) / sqrt[(263362 - (495.2)^2)(52372.6 - (227.6)^2)]

r = 2671.38 / sqrt[(26336.2 - 245.04)(5237.26 - 51.76)]

r = 2671.38 / sqrt[26191.16 * 5185.5]

r = 2671.38 / sqrt[135786357.38]

r ≈ 0.685

Therefore, the correlation coefficient between milk produced and cattle raised is approximately 0.685.