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.
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
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.
To investigate the relationship between milk produced (x) and cattle raised (y), we can use linear regression analysis. Linear regression analysis helps us determine the relationship between two variables and predict values based on that relationship.
To calculate the regression line, we need to find the slope (b) and the y-intercept (a).
The slope (b) of the regression line can be calculated using the following formula:
b = (n ∑ xy - ∑ x ∑ y) / (n ∑ x^2 - (∑ x)^2)
where n is the number of data points.
To calculate the y-intercept (a), we can use the following formula:
a = (∑ y - b ∑ x) / n
Let's calculate the slope (b) and y-intercept (a) using the given information:
n = 10 (number of data points)
∑ x = 495.2
∑ y = 227.6
∑ x^2 = 26336.2
∑ y^2 = 5237.26
∑ xy = 11543.89
Now, let's calculate the slope (b):
b = (n ∑ xy - ∑ x ∑ y) / (n ∑ x^2 - (∑ x)^2)
= (10 * 11543.89 - 495.2 * 227.6) / (10 * 26336.2 - 495.2^2)
= (115438.9 - 112772.32) / (26336.2 - 245.04)
= 2666.58 / 26091.16
≈ 0.102
The slope of the regression line is approximately 0.102.
Now, let's calculate the y-intercept (a):
a = (∑ y - b ∑ x) / n
= (227.6 - 0.102 * 495.2) / 10
= (227.6 - 50.4644) / 10
= 177.13 / 10
≈ 17.713
The y-intercept of the regression line is approximately 17.713.
Therefore, the equation of the regression line is y ≈ 0.102x + 17.713.
This equation represents the relationship between milk produced (x) and cattle raised (y). To predict the amount of milk produced based on the number of cattle raised, you can substitute the value of x into the equation and calculate y.
To investigate the relationship between milk produced (x) and cattle raised (y) in South Africa, we can use various statistical methods. One common approach is to calculate the linear regression equation using the given data.
Step 1: Calculate the mean values for x and y.
Mean of x (x̄) = ∑x / n = 495.2 / 10 = 49.52
Mean of y (ȳ) = ∑y / n = 227.6 / 10 = 22.76
Step 2: Calculate the deviations from the mean for both x and y.
Deviation of x (dx) = xi - x̄
Deviation of y (dy) = yi - ȳ
The deviations from the mean for x are:
-20.82, -13.22, -8.22, -8.42, -3.12, -1.82, 5.18, 7.38, 21.68, 22.28
The deviations from the mean for y are:
-5.06, -1.36, -1.56, 0.94, 1.04, -0.46, -0.46, -0.86, 2.44, 4.34
Step 3: Calculate the sum of squared deviations (SSD) for both x and y.
SSDx = ∑(dx^2) = ∑((xi - x̄)^2) = ∑(xi^2) - n(x̄^2) = ∑x^2 - n(x̄^2) = 26336.2 - (10)(49.52^2) = 26336.2 - 24615.2048 = 1720.9952
SSDy = ∑(dy^2) = ∑((yi - ȳ)^2) = ∑(yi^2) - n(ȳ^2) = ∑y^2 - n(ȳ^2) = 5237.26 - (10)(22.76^2) = 5237.26 - 5203.9776 = 33.2824
Step 4: Calculate the sum of cross-products of deviations (SPD) for x and y.
SPD = ∑(dx * dy) = ∑((xi - x̄)(yi - ȳ)) = ∑(xi * yi) - n(x̄ * ȳ) = ∑xy - n(x̄ * ȳ) = 11543.89 - (10)(49.52)(22.76) = 11543.89 - 11352.336 = 191.554
Step 5: Calculate the slope of the regression line (b).
b = SPD / SSDx = 191.554 / 1720.9952 = 0.1114
Step 6: Calculate the y-intercept of the regression line (a).
a = ȳ - b * x̄ = 22.76 - (0.1114)(49.52) = 22.76 - 5.509728 = 17.250272
Step 7: Write the equation of the regression line.
y = a + b * x
y = 17.250272 + 0.1114 * x
Therefore, the equation for the linear regression line is y = 17.250272 + 0.1114x.