Sketch a scatter diagram that might be expected when $x$ and $y$ are related approximately

Question

Sketch a scatter diagram that might be expected when $x$ and $y$ are related approximately as given in each of the cases (A) and (B) below. In each case, your diagram should include $6$ points, approximately equally spaced with respective to $x$, and with all $x$- and $y$-values positive. The letters $a$, $b$, $c$ and $d$ represent constants. $y=a+b\sqrt{x}$, where $a$ is positive and $b$ is negative $y=c+\frac{\mathrm{d}}{x}$, where $c$ is positive and $d$ is negative. The data below shows the diameter $h$, in cm, of coconuts of $9$ different ages, in $k$ months. Age in month ($k$)123.54.557.5910.512 Diameter in cm ($h$)69.5131516.517.518.51919.5 Calculate the value of the product moment correlation coefficient $r$ between $k$ and $h$. State, with a reason, whether the value of $r$ would be different if the diameter is measured in inches instead.[2] Give a sketch of the scatter diagram for the data.[1] It is desired to predict the diameter of coconuts in the long run. Explain why neither a linear nor a quadratic model is likely to be appropriate.[2] It is suggested that diameter $h$ can be modelled by the formula $h=m+n\ln k$. Find the equation of the regression line for the suggested model, and the value of the product moment correlation coefficient between $\ln k$ and $h$.[2] Estimate, to $1$ decimal place, the age of a coconut with a diameter of $13.7$ cm, and comment on the reliability of your answer.[2] Given that $1$ inch $=2.54$cm, state the equation of regression line if the diameter is measured in inches instead of cm.[1]

Accepted Answer

$r = 0.911$; $h = 6.15 + 5.61\ln k$, $r = 0.992$; $k = 3.8$ months; $h = 2.21\ln k + 2.42$

2023 DHS P2 Q11

2023 DHS P2 Q11

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