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

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
 
You have to test the hypothesis that there is no correlation between milk and
cattle raised. The value of the test statistic is

GPT 3.5 · Answer

To test the hypothesis that there is no correlation between milk produced and cattle raised, we can use the formula for the test statistic:

r = ∑xy - (∑x)(∑y)/n
_____________________
√(∑x^2 - (∑x)^2/n)(∑y^2 - (∑y)^2/n)

where r is the sample correlation coefficient, ∑xy is the sum of the products of x and y, ∑x is the sum of x values, ∑y is the sum of y values, ∑x^2 is the sum of squared x values, ∑y^2 is the sum of squared y values, and n is the number of observations.

Using the given values:

∑x = 495.2
∑y = 227.6
∑x^2 = 26336.2
∑y^2 = 5237.26
∑xy = 11543.89
n = 10

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

r = (11543.89 - (495.2)(227.6)/10) / (√((26336.2 - (495.2)^2/10)(5237.26 - (227.6)^2/10)))

r = (11543.89 - 112630.72/10) / (√((26336.2 - 24520.24/10)(5237.26 - 51910.24/10)))

r = (11543.89 - 11263.07) / (√((26336.2 - 2452.02)(5237.26 - 519.10)))

r = 280.82 / (√(23884.18 * 4718.16))

r = 280.82 / (√112795207.9728)

r = 280.82 / 10629.92

r ≈ 0.0265

The sample correlation coefficient is approximately 0.0265.

To test the hypothesis, we can compare the absolute value of the correlation coefficient (|r|) to a critical value. Let's assume a significance level of 0.05. The critical value for a two-tailed test with n=10 is approximately 0.685.

Since |0.0265| < 0.685, we fail to reject the null hypothesis. Therefore, there is not enough evidence to suggest a significant correlation between milk produced and cattle raised.