In an experiment, different quantities of fertilizer, $x$ ml, were given to seven radish plants of

Question

In an experiment, different quantities of fertilizer, $x$ ml, were given to seven radish plants of the same height. The heights, $y$ mm, of the plants were measured after ten days. The results are given in the table. $x$ $10$ $15$ $20$ $25$ $30$ $35$ $40$ $y$ $407$ $412$ $420$ $434$ $450$ $465$ $490$ Draw a scatter diagram to illustrate the data. [1] It is suggested that the height $y$ can be modelled by one of the formulae $y=a+bx$ or $\text{}\hspace{0.5em}y=c+d{{x}^{2}}$ where $a$, $b$, $c$ and $d$ are constants. Find, correct to 4 decimal places, the value of the product moment correlation coefficient between $x$ and $y$, ${{x}^{2}}$and $y$.[2] Use your answers to parts (i) and (ii) to explain which of $y=a+bx$ or $y=c+d{{x}^{2}}$ is the better model.[1] It is desired to estimate the value of $x$ for which $y=440.$ Explain why neither the regression line of $x$ on $y$ nor the regression line of ${{x}^{2}}$ on $y$ should be used.[1] By finding the equation of a suitable regression line, find the required estimate in part (iv). Comment on the reliability of your estimate.[3]

Accepted Answer

(ii)(a) $r=0.9781$ ($4$ d.p.) (b) $r=0.9985$($4$ d.p.) (iii) From part (i), the values of $y$are increasing at an increasing rate or the points concave upwards which is similar to that of a quadratic graph. From part (ii), the $r$-value between ${{x}^{2}}$and $y$ is closer to 1 than the $r$-value between $x$ and $y$ indicating a stronger positive linear relationship between ${{x}^{2}}$and $y$. $	herefore 	ext{ }y=c+d{{x}^{2}}$ is the better model. (iv) As $y$ is the dependent variable and $x$ is the independent (controlled) variable, neither the regression line of $x$ on $y$ nor the regression line of ${{x}^{2}}$ on $y$ should be used. (v) $y=399.60+0.055326{{x}^{2}}$, $x=27.0$; The estimate is reliable as there is a strong positive linear relationship between ${{x}^{2}}$and $y$ and interpolation is used, i.e. the value of $y$ is within the data range.

2015 DHS P2 Q10

2015 DHS P2 Q10