2021 H2 Maths Paper 2 Q9 — Worked Solution

Part (a)

1 mark

No — a linear model is not likely to fit well.

Explanation: although $r = 0.9803$ is close to $1$ , indicating a strong linear association, the scatter diagram shows the points bending upward as $t$ increases, a curved pattern rather than a straight line. A high value of $r$ measures only the strength of a linear relationship; it does not rule out a non-linear model fitting even better, so the curvature visible in the diagram means a straight line is not a good model here.

$\therefore$ a linear model is not likely to be a good model for $x$ against $t$ .

Part (b)(i)

1 mark

We compute $\ln x$ for each value of $x$ in the table and plot it against $t$ .

$\therefore$ the points now lie close to a straight line with positive gradient, with none of the curvature seen in the original $x$ against $t$ diagram.

Part (b)(ii)

3 marks

By GC, performing linear regression of $\ln x$ on $t$ ,

\ln x = c + dt

c \approx 2.29,\quad d \approx 0.0573\ (3\ \text{s.f., by GC)}

and the product moment correlation coefficient for this model is

r \approx 0.999\ (3\ \text{s.f., by GC)}.

To check your answer: at

t = 24

\ln x \approx 2.293 + 0.0573(24) = 3.68

, so

x \approx e^{3.68} \approx 39.6

, close to the given

x = 39.3

. ✓

$\therefore$ $c \approx 2.29$ , $d \approx 0.0573$ , and $r \approx 0.999$ (3 s.f.).

Part (c)

2 marks

The model $\ln x = c + dt$ has $r \approx 0.999$ , closer to $1$ than the linear model’s $r = 0.9803$ , and its scatter diagram in (b)(i) shows the points lying close to a straight line with no visible curvature, unlike the original diagram. Since $\ln x = c + dt$ is equivalent to $x = e^{c}e^{dt} = Ae^{dt}$ (an exponential growth model, where $A = e^{c}$ ), Toby’s second model fits the data much better than his first, straight-line model for $x$ against $t$ .

$\therefore$ the exponential model $x = Ae^{dt}$ , obtained from $\ln x = c + dt$ , is a much better fit than the straight-line model $x$ against $t$ .

2021 H2 Maths Paper 2, Question 9

Part (a)

Part (b)(i)

Part (b)(ii)

Part (c)

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