2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species.
| Age \(( x\) years \()\) | 11 | 12 | 20 | 28 | 35 | 45 | 51 |
| Mean trunk diameter \(( y \mathrm {~cm} )\) | 12.2 | 16.0 | 26.4 | 39.2 | 39.6 | 51.3 | 60.6 |
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
- (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
(b) Find the equation of the regression line of \(y\) on \(x\). - Use your equation to estimate the mean trunk diameter of a tree of this species with age
(a) 30 years,
(b) 100 years.
It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places. - Comment on the reliability of each of your two estimates.