A batch of plants is grown and then studied to examine how well they are growing. For each plant, da

Question

A batch of plants is grown and then studied to examine how well they are growing. For each plant, data on two indicators are collected: the average root hair length $x$ mm, and the amount of potassium found in the roots $y$ mg. The data from a random sample of $8$ plants are shown in the table below.

Average root hair

length

($x$ mm)

$3.3$

$3.9$

$7.5$

$8.1$

$13.3$

$22.1$

$32.2$

$36.1$

Amount of

potassium

($y$ mg)

$9$

$t$

$25$

$27$

$33$

$37$

$41$

$44$

1. Calculate an unbiased estimate of the population variance of the average root hair lengths.[1]
2. Find an expression for the unbiased estimate of the population variance of the amount of potassium in the roots, in the form
  $\frac{1}{A}\left[ \left( B+{{t}^{2}} \right)-\frac{{{\left( C+t \right)}^{2}}}{D} \right]$,
  where $A$, $B$, $C$ and $D$ are positive integers to be determined.[2]

You are now given that $t=17$.

Sketch a scatter diagram of $y$ against $x$. State the value of the product moment correlation coefficient between $y$ and $x$.[2]
With reference to your answers in part (b), explain whether an equation of the form $y=mx+k$, where $m$ and $k$ are constants, provides an accurate model of the relationship between $x$ and $y$.[1]
The data is instead modelled by the regression equation $y=p\ln x+q$. Find the values of $p$ and $q$, and the value of the corresponding product moment correlation coefficient.[2]
Give a contextual interpretation for the value of $p$ found in part (d).[1]
Hence find the estimated amount of potassium in the roots of a plant with average root hair length $30.0$mm. Explain whether your estimate is reliable.[2]
It is now required to estimate the average root hair length for a plant with $28$ mg of potassium in its roots. Explain whether it is appropriate to use the equation in part (d) to obtain this estimate.[1]

Accepted Answer

(a)(i) ${165}$ (ii) ${A = 7}$, ${B = 7510}$, ${C = 216}$ and ${D = 8}$. (c) No, from the sketch there is a non-linear relationship between the variables. (d) ${p = 12.9}$, ${q = -2.27}$ (${3}$ s.f.), ${r = 0.984}$. (e) For every unit increase in the log average hair root length, the amount of potassium found in the roots increases by ${12.9	ext{ mg}}$. (f) ${41.8	ext{ mg}}$. Yes it is reliable because ${\left\vert{}right\vert{}}$ is close to ${1}$ showing a strong linear relationship between ${y}$ and ${\ln x}$, and the value of ${30.0	ext{ mm}}$ lies within the range of values of ${x}$ (${3.3 < x < 36.1}$). (g) No, because there is no clear independent & dependent variable, so a regression line of ${\ln x}$ on ${y}$ would be more appropriate

2023 EJC P2 Q10

2023 EJC P2 Q10

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