Correlation & Regression: Regression; interpolate — EJC 2025 H2 Math Prelim Paper 2
What this question tests
Question
An investigation into the relationship between two variables \(x\) and \(y\) results in the following data.
| \(x\) | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|
| \(y\) | 1.1 | 5.7 | 9.3 | 9.8 | 8.6 | 6.4 | 0.9 |
- Calculate the product moment correlation coefficient between \(x\) and \(y\).
- Draw a scatter diagram of the data and comment on the relationship between \(x\) and \(y\) based on the scatter diagram.
Several possible models for the relationship between \(x\) and \(y\) are proposed; one is chosen for further investigation.
- Calculate the product moment correlation coefficient between \(y\) and \((x - 10)^2\), and comment on its value.
- Find the equation of the regression line of \(y\) on \((x - 10)^2\). Use the equation of the regression line to estimate the value of \(y\) when \(x = 17\) and comment on the reliability of this estimate.
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(a) From the GC, the product moment correlation coefficient between \(x\) and \(y\) is \[ r = 0.00208 \quad (\text{3 s.f.}). \]
\(r = 0.00208\).
(b) Scatter diagram:
The points rise to a peak around \(x = 10\) and fall on either side — there is a clear non-linear (curved) relationship between \(x\) and \(y\), consistent with the very low linear correlation in (a).
There is a non-linear (curved) relationship between \(x\) and \(y\).
(c) From the GC, the product moment correlation coefficient between \(y\) and \((x-10)^2\) is \[ r = -0.997 \quad (\text{3 s.f.}). \] This value is very close to \(-1\), indicating a very strong negative linear correlation between \(y\) and \((x-10)^2\).
\(r = -0.997\). Very strong negative linear correlation between \(y\) and \((x-10)^2\).
(d) From the GC, the regression line of \(y\) on \((x-10)^2\) is \[ y = -0.987(x-10)^2 + 9.92. \]
When \(x = 17\), \((x-10)^2 = 49\), so \[ y = -0.987(49) + 9.92 = -38.4 \quad (\text{3 s.f.}). \]
This estimate is not reliable, because \(x = 17\) lies outside the data range \(7 \le x \le 13\) (extrapolation), so the linear relationship between \(y\) and \((x-10)^2\) is not guaranteed to hold there.