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The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of \(y\) on \(x\). State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram.
State, giving a reason, whether, for the data shown, the regression line of \(y\) on \(x\) is the same as the regression line of \(x\) on \(y\).
A car is travelling along a stretch of road with speed \(v \mathrm {~km} \mathrm {~h} ^ { - 1 }\) when the brakes are applied. The car comes to rest after travelling a further distance of \(z \mathrm {~m}\). The values of \(z\) (and \(\sqrt { } z\) ) for 8 different values of \(v\) are given in the table, correct to 2 decimal places.
| \(v\) | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| \(z\) | 2.83 | 4.63 | 4.84 | 5.29 | 9.73 | 10.30 | 14.82 | 15.21 |
| \(\sqrt { } z\) | 1.68 | 2.15 | 2.20 | 2.30 | 3.12 | 3.21 | 3.85 | 3.90 |
$$\left[ \Sigma v = 340 , \Sigma v ^ { 2 } = 15500 , \Sigma \sqrt { } z = 22.41 , \Sigma z = 67.65 , \Sigma v \sqrt { } z = 1022.15 . \right]$$
- Calculate the product moment correlation coefficient between \(v\) and \(\sqrt { } z\). What does this indicate about the scatter diagram of the points \(( v , \sqrt { } z )\) ?
- Given that the product moment correlation coefficient between \(v\) and \(z\) is 0.965 , correct to 3 decimal places, state why the regression line of \(\sqrt { } z\) on \(v\) is more suitable than the regression line of \(z\) on \(v\), and find the equation of the regression line of \(\sqrt { } z\) on \(v\).
- Comment, in the context of the question, on the value of the constant term in the equation of the regression line of \(\sqrt { } z\) on \(v\).