Draw separate scatter diagrams, each with $7$ points, all in the first quadrant, which represent t

Question

Draw separate scatter diagrams, each with $7$ points, all in the first quadrant, which represent the situation where the product moment correlation coefficient between variables $x$ and $y$ is
1. $0$,[1]
2. approximately $0.8$.[1]
Eight cities in a certain country are linked by rail to the capital city. The table below shows the distance of each city from the capital and the rail price from the city to the capital.
City $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$
Distance, $d\text{ km}$ $124$ $44$ $76$ $148$ $16$ $180$ $104$ $195$
Rail price, $\$p$ $156$ $53$ $99$ $169$ $23$ $177$ $138$ $178$
1. Draw a scatter diagram of the data.
  [2]
2. For a line of best fit $p = \mathrm{f}(d)$, the residual for a point $(a, b)$ plotted on the scatter diagram is the vertical distances between $(a, \mathrm{f}(a))$ and $(a, b)$. Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.
  [1]
3. Using the scatter diagram in (i), explain if a linear relationship between the distance and the rail price is appropriate.
  [1]
4. Suppose that the relationship between $d$ and $p$ are modelled by an equation of the form $p = s + t \ln d$, where $s$ and $t$ are constants. Find the equation of this regression line and use it to estimate the rail price for a city which is $210\text{ km}$ from the capital.
  [2]
5. Comment on the reliability of the estimate in (iv).
  [1]
6. Following an adjustment to the rail prices, all rail prices are increased by $10$ dollars. Without any further calculations, state any change you would expect in the values of the constants $s$ and $t$ found in (iv).
  [2]

Accepted Answer

(a)(i) $r=0$ (ii) one possible diagram has $r=0.813$ (b)(iv) $p=-184.595+68.9582\ln d$, giving $p=184.13$ at $d=210$ (v) unreliable extrapolation (vi) $s$ increases by $10$, while $t$ is unchanged

City	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Distance, \(d\text{ km}\)	\(124\)	\(44\)	\(76\)	\(148\)	\(16\)	\(180\)	\(104\)	\(195\)
Rail price, \(\$p\)	\(156\)	\(53\)	\(99\)	\(169\)	\(23\)	\(177\)	\(138\)	\(178\)

2025 JPJC P2 Q11

2025 JPJC P2 Q11

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