Compare correlation coefficients

3 A sample of bivariate data was taken and the results were summarised as follows. $$n = 5 \quad \Sigma x = 24 \quad \Sigma x ^ { 2 } = 130 \quad \Sigma y = 39 \quad \Sigma y ^ { 2 } = 361 \quad \Sigma x y = 212$$

1. Show that the value of the product moment correlation coefficient \(r\) is 0.855 , correct to 3 significant figures.
2. The ranks of the data were found. One student calculated Spearman's rank correlation coefficient \(r _ { s }\), and found that \(r _ { s } = 0.7\). Another student calculated the product moment coefficient, \(R\), of these ranks. State which one of the following statements is true, and explain your answer briefly.
   (A) \(R = 0.855\)
   (B) \(R = 0.7\)
   (C) It is impossible to give the value of \(R\) without carrying out a calculation using the original data.
3. All the values of \(x\) are now multiplied by a scaling factor of 2 . State the new values of \(r\) and \(r _ { s }\).

4 The girth, \(g\) metres, the length, \(l\) metres, and the weight, \(y\) kilograms, of each of a sample of 20 pigs were measured. The data collected is summarised as follows. $$S _ { g g } = 0.1196 \quad S _ { l l } = 0.0436 \quad S _ { y y } = 5880 \quad S _ { g y } = 24.15 \quad S _ { l y } = 10.25$$

1. Calculate the value of the product moment correlation coefficient between:
   1. girth and weight;
   2. length and weight.
2. Interpret, in context, each of the values that you obtained in part (a).
3. Weighing pigs requires expensive equipment, whereas measuring their girths and lengths simply requires a tape measure. With this in mind, the following formula is proposed to make an estimate of a pig's weight, \(x\) kilograms, from its girth and length. $$x = 69.3 \times g ^ { 2 } \times l$$ Applying this formula to the relevant data on the 20 pigs resulted in $$S _ { x x } = 5656.15 \quad S _ { x y } = 5662.97$$
   1. By calculating a third value of the product moment correlation coefficient, state which of \(g , l\) or \(x\) is the most strongly correlated with \(y\), the weight.
   2. Estimate the weight of a pig that has a girth of 1.25 metres and a length of 1.15 metres.
   3. Given the additional information that \(\bar { x } = 115.4\) and \(\bar { y } = 116.0\), calculate the equation of the least squares regression line of \(y\) on \(x\), in the form \(y = a + b x\).
   4. Comment on the likely accuracy of the estimated weight found in part (c)(ii). Your answer should make reference to the value of the product moment correlation coefficient found in part (c)(i) and to the values of \(b\) and \(a\) found in part (c)(iii).
      (4 marks)