2016 H2 Maths Paper 2, Question 8

The circled point is $(1.5,\ 82.5)$, which breaks the otherwise smooth increasing pattern of the data.

Explanation: the scatter diagram shows the points rising steeply at first and then levelling off towards a roughly horizontal spread as $x$ increases, i.e. the relationship is curved (concave), not a straight line. A model $y = ax + b$ would give a constant rate of increase in $y$ for a fixed increase in $x$, which does not match the way the points bend over for large $x$.

Since $y$ increases as $x$ increases (efficiency rises with power), we need $\frac{dy}{dx} > 0$.

For $\frac{dy}{dx} > 0$ we need $- c > 0$, i.e. $c < 0$.

As $x \rightarrow \infty$, $\frac{c}{x} \rightarrow 0$, so $y \rightarrow d$. Since the efficiency values level off at a positive percentage (around $90\%$–$92\%$) rather than at $0$ or below, $d > 0$.

Let $u = \frac{1}{x}$. Excluding $x = 1.5$, the data become

By GC, the regression line of $y$ on $u$ is

so $c = - 18.5$ and $d = 91.9$, both correct to 3 s.f.

Using the model with $x = 1.5$,

$\therefore$ the estimated efficiency value that Xian copied wrongly is $79.6\%$.

Explanation: this estimate is expected to be reliable for two reasons. First, $x = 1.5$ lies within the range of $x$-values ($1 \leq x \leq 50$) used to obtain the model, so this is interpolation, not extrapolation. Second, the value of $r = - 0.986$ found in part (iv) is very close to $- 1$, showing that $y$ and $u = \frac{1}{x}$ are very strongly (linearly) correlated, so the model fits the remaining data closely.