N2014 P2 Q8

Junior College 2

8 marks

Sketch a scatter diagram that might be expected when $x$ and $y$ are related approximately by $y=px^2+t$ in each of the cases (i) and (ii) below. In each case your diagram should include $6$ points, approximately equally spaced with respect to $x$, and with all $x$-values positive.[2]
1. $p$ and $t$ are both positive.
2. $p$ is negative and $t$ is positive.
The ages in months $m$ and prices in dollars $P$ of a random sample of ten used cars of a certain model are given in the table.
$m$ $11$ $20$ $28$ $36$ $40$ $47$ $58$ $62$ $68$ $75$
$P$ $112800$ $102600$ $76500$ $72000$ $72000$ $69000$ $65800$ $57000$ $50600$ $47600$
It is thought that the relationship between $m$ and $P$ can be modelled by either $P=am+b$ or $P=c\ln m+d$.
1. Find, correct to $4$ decimal places, the product moment correlation coefficient between (A) $m$ and $P$, (B) $\ln m$ and $P$.[2]
2. Explain which model is more appropriate and find the equation of the corresponding regression line.[3]
3. Use your chosen model to estimate the price of a car which is $50$ months old.[1]

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Answer:(b)(i)(A) $-0.9470$, (B) $-0.9749$. (ii) $P=195693.5593\ldots-33659.72805\ldots\ln m$. (iii) Approximately $\$64\,000$.

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\(m\)	\(11\)	\(20\)	\(28\)	\(36\)	\(40\)	\(47\)	\(58\)	\(62\)	\(68\)	\(75\)
\(P\)	\(112800\)	\(102600\)	\(76500\)	\(72000\)	\(72000\)	\(69000\)	\(65800\)	\(57000\)	\(50600\)	\(47600\)