N2017 P2 Q8

Junior College 2

10 marks

Draw separate scatter diagrams, each with $8$ points, all in the first quadrant, which represent the situation where the product moment correlation coefficient between variables $x$ and $y$ is
1. $-1$,
2. $0$,
3. between $0.5$ and $0.9$.[3]
An investigation into the effect of a fertiliser on yields of corn found that the amount of fertiliser applied, $x$, resulted in the average yields of corn, $y$, given below, where $x$ and $y$ are measured in suitable units.
$x$ $0$ $40$ $80$ $120$ $160$ $200$
$y$ $70$ $104$ $118$ $119$ $126$ $129$
1. Draw a scatter diagram for these values. State which of the following equations, where $a$ and $b$ are positive constants, provides the most accurate model of the relationship between $x$ and $y$.
  (A) $y=ax^2+b$; (B) $y=\dfrac a{x^2}+b$; (C) $y=a\ln2x+b$; (D) $y=a\sqrt x+b$.[2]
2. Using the model you chose in part (i), write down the equation for the relationship between $x$ and $y$, giving the numerical values of the coefficients. State the product moment correlation coefficient for this model.[3]
3. Give two reasons why it would be reasonable to use your model to estimate the value of $y$ when $x=189$.[2]

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Answer:(a) See example scatter diagrams; (b)(i) D; (ii) $y=4.18\sqrt x+74.0$, $r=0.981$; (iii) interpolation and strong transformed correlation.

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\(x\)	\(0\)	\(40\)	\(80\)	\(120\)	\(160\)	\(200\)
\(y\)	\(70\)	\(104\)	\(118\)	\(119\)	\(126\)	\(129\)