3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.
| \(x\) | 148 | 164 | 156 | 172 | 147 | 184 | 162 | 155 | 182 | 165 | 175 | 152 |
| \(y\) | 39 | 59 | 56 | 77 | 44 | 77 | 65 | 49 | 80 | 72 | 70 | 52 |
- On graph paper, draw a scatter diagram to represent these data.
- Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative.
The data in the table can be summarised as follows.
$$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
- Find \(S _ { x y }\).
The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
- Find, to 3 decimal places, the value of \(b\).
- Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
- Find the values of \(\bar { y } \pm 1.96 s\).
- Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.