2 A medical student is trying to estimate the birth weight of babies using pre-natal scan images. The actual weights, \(x \mathrm {~kg}\), and the estimated weights, \(y \mathrm {~kg}\), of ten randomly selected babies are given in the table below.
| \(x\) | 2.61 | 2.73 | 2.87 | 2.96 | 3.05 | 3.14 | 3.17 | 3.24 | 3.76 | 4.10 |
| \(y\) | 3.2 | 2.6 | 3.5 | 3.1 | 2.8 | 2.7 | 3.4 | 3.3 | 4.4 | 4.1 |
- Calculate the value of Spearman's rank correlation coefficient.
- Carry out a hypothesis test at the \(5 \%\) level to determine whether there is positive association between the student's estimates and the actual birth weights of babies in the underlying population.
- Calculate the value of the product moment correlation coefficient of the sample. You may use the following summary statistics in your calculations:
$$\Sigma x = 31.63 , \quad \Sigma y = 33.1 , \quad \Sigma x ^ { 2 } = 101.92 , \quad \Sigma y ^ { 2 } = 112.61 , \quad \Sigma x y = 106.51 .$$
- Explain why, if the underlying population has a bivariate Normal distribution, it would be preferable to carry out a hypothesis test based on the product moment correlation coefficient.
Comment briefly on the significance of the product moment correlation coefficient in relation to that of Spearman’s rank correlation coefficient.