RI 2025 H2 Math Prelim Paper 2 — Correlation & Regression (worked solution)

Correlation & Regression: Regression + PMCC + estimate — RI 2025 H2 Math Prelim Paper 2

Raffles Institution2025 PrelimPaper 2●●○ Standard10 marks

What this question tests

Regression + PMCC + estimate.

Question

A small café sells one type of coffee. The selling price of a cup of coffee is reviewed and adjusted at the beginning of every year depending on market conditions. Based on sales figures collected over 7 years, the café owner, Mr Tay, studied the effect of the selling price of a cup of coffee, \$x, on the average number of cups, $y$ cups, sold per day within the year. The data is shown in the table below.

$x$	2.0	2.2	2.5	3.0	4.0	4.4	4.5
$y$	280	250	190	150	90	100	70

On the grid in the Printed Answer Booklet, draw a scatter diagram of the data.

Mr Tay realised that one of the values of $y$ has been wrongly stated in the data table. Indicate the corresponding point on your diagram by labelling it $P$. Explain why the scatter diagram for the remaining points may be consistent with a model of the form $y = a + \dfrac{b}{x}$, where $a$ and $b$ are constants.

Omitting $P$, calculate the product moment correlation coefficient and the least squares estimates of $a$ and $b$ for the model $y = a + \dfrac{b}{x}$.

Use the model $y = a + \dfrac{b}{x}$, with the values of $a$ and $b$ found in part (c), to estimate the value of $y$ that has been wrongly stated in the data table. Give two reasons why you would expect this estimate to be reliable.

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Diagram for RI 2025 P2

As $x$ increases, $y$ decreases but with decreasing amounts (the rate of decrease slows). This is consistent with $y = a + \dfrac{b}{x}$ (with $b < 0$), which decreases at a decreasing rate.

The anomalous point is $P(4.4, 100)$ — it appears too high compared with the trend.

Omitting $P$ (i.e. omitting the point $(4.4, 100)$), using $x' = \dfrac{1}{x}$:

The 6 remaining data points give: \[\begin{aligned} r &= 0.998 \text{ (3\,s.f.)}\\[4pt] a &= -101 \text{ (3\,s.f.)}\\[4pt] b &= 757 \text{ (3\,s.f.)} \end{aligned}\]

Using $y = -101 + \dfrac{757}{x}$ at $x = 4.4$: \[\begin{aligned} y &= -101 + \frac{757}{4.4} = -101 + 172.05\ldots \approx 71.2 \end{aligned}\]

This estimate is reliable because:

$r = 0.998 \approx 1$, indicating a very strong positive linear correlation between $y$ and $\dfrac{1}{x}$, suggesting the model $y = a + \dfrac{b}{x}$ is appropriate.
$x = 4.4$ lies within the range of given data $[2.0, 4.5]$, so this is interpolation rather than extrapolation.

Answer: (c) $r = 0.998$, $a = -101$, $b = 757$ (d) Estimated value: $y \approx 71.2$ (i.e. the correct data value should be approximately 71).

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