John is training for National Shooting Championship and he practices shooting each week. His percent

Question

John is training for National Shooting Championship and he practices shooting each week. His percentage score, $y\%$ in week $x$, are as follows.

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$y$	$40$	$57$	$68$	$74$	$80$	$85$	$88$

Draw a scatter diagram to illustrate the data.[2]
Calculate the value of the product moment correlation coefficient, and explain why its value does not necessarily mean that a linear model is an appropriate model for the relationship between $x$ and $y$.[2]

It is decided to fit a model of the form $\mathrm{ln}\left( 95-y \right)=a+bx$.

Find, correct to $4$ decimal places, the value of the product moment correlation coefficient between $\mathrm{ln}\left( 95-y \right)$ and $x$ and comment on its value.[2]
Calculate the values of $a$ and $b$ correct to $4$ decimal places. Use them to predict, to the nearest whole number, the percentage score John will obtain for week $8$. Comment on the reliability of the estimate.[4]
Give an interpretation, in context, of the value $95$ in the model $\mathrm{ln}\left( 95-y \right)=a+bx$.[1]

Accepted Answer

(i) Scatter diagram plotted (ii) $r=0.961$; the curved, bounded context means a linear model need not be appropriate (iii) $r=-0.9986$, indicating an almost perfect negative linear relationship (iv) $a=4.3406$, $b=-0.3372$, predicted week-8 score $90\%$; less reliable because it is extrapolation (v) $95\%$ is the limiting score John approaches after many weeks

2013 IJC P2 Q9

2013 IJC P2 Q9

Video Solution:

Solution:

Solution locked