The data below shows the average height, measured in feet, of cherry trees from age $1$ year to ag

Question

The data below shows the average height, measured in feet, of cherry trees from age $1$ year to age $11$ years.

Age ($x$)	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
Average height ($y$)	$6$	$9.5$	$13$	$15$	$16.5$	$17.5$	$18.5$	$19$	$19.5$	$19.7$	$19.8$

Calculate $r$, the value of the product moment correlation coefficient between $x$ and $y$. State, with a reason, whether the value of $r$ would be different if the average height is measured in meters instead.[2]
Give a sketch of the scatter diagram for the data.[1]
It is desired to predict the average height of cheery trees beyond age $11$ years. Explain why neither a linear nor a quadratic model is likely to be appropriate.[2]

It is suggested that the average height $y$ can be modelled by the formula $y=a+b\ln x$.

Find the equation of the regression line for the suggested model, and the product moment correlation coefficient between $\ln x$ and $y$. Estimate the average height of cherry trees age $5.5$ years, and comment on the reliability of your answer.[4]

Accepted Answer

(i) $r=0.916$; unchanged when height is measured in metres (ii) Scatter diagram plotted (iii) Neither a linear nor a quadratic model suitably represents the levelling trend beyond age $11$ (iv) $y=6.10818\ln x+6.09934$, $r=0.993$; at $x=5.5$, $y=16.5$ feet, a reliable interpolation

2013 RI P2 Q9

2013 RI P2 Q9

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