<ol class="abc-list"><li><div class="list-item-paragraph">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.</div><div class="list-item-paragraph" style="text-align: center; margin-left: 1rem">(A) $y=a+b\sqrt{x}$, where $a$ is positive and $b$ is negative</div><div class="list-item-paragraph" style="text-align: center">(B) $y=c+\frac{\\mathrm{d}}{x}$, where $c$ is positive and $d$ is negative.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The data below shows the diameter $h$, in cm, of coconuts of $9$ different ages, in $k$ months.</div><table><thead><tr><th>Age in month ($k$)</th><th>1</th><th>2</th><th>3.5</th><th>4.5</th><th>5</th><th>7.5</th><th>9</th><th>10.5</th><th>12</th></tr></thead><tbody><tr><td>Diameter in cm ($h$)</td><td>6</td><td>9.5</td><td>13</td><td>15</td><td>16.5</td><td>17.5</td><td>18.5</td><td>19</td><td>19.5</td></tr></tbody></table><ol class="i-list"><li><div class="list-item-paragraph">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.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Give a sketch of the scatter diagram for the data.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">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.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">It is suggested that diameter $h$ can be modelled by the formula $h=m+n\ln k$.</div></li><li><div class="list-item-paragraph">Find the equation of the regression line for the suggested model, and the value of the product moment correlation coefficient between $\ln k$ and $h$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">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.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $1$ inch $=2.54$cm, state the equation of regression line if the diameter is measured in inches instead of cm.</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

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.

(A) $y=a+b\sqrt{x}$, where $a$ is positive and $b$ is negative

(B) $y=c+\frac{\mathrm{d}}{x}$, 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.

2023 DHS P2 Q11

2023 DHS P2 Q11

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