Correlation & Regression: PMCC; prediction — VJC 2025 H2 Math Prelim Paper 2
What this question tests
Question
An energy company is studying the relationship between the average daily temperature, \(x\) (in degrees Celsius), and the amount of electricity consumed, \(y\) (in megawatt-hours MWh). For a random sample of ten days, the records are shown in the table below.
| Average daily temperature (\(x\,^{\circ}\)C) | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 |
|---|---|---|---|---|---|---|---|---|---|---|
| Electricity consumption (\(y\) MWh) | 12.5 | 12.8 | 13.2 | 14.0 | 14.7 | 16.2 | 16.2 | 19.9 | 22.3 | 29.5 |
- (a) Draw a scatter diagram for these values, labelling the axes clearly.
- (b) Find the value of the product moment correlation coefficient between
- (i) \(x\) and \(y\),
- (ii) \(x^2\) and \(y\).
- (c) Using your answers in parts (a) and (b), explain which of \(y = a + bx^2\) and \(y = c + dx\) is the better model and find the equation of a suitable regression line for this model.
- (d) Use the equation of your regression line to estimate the electricity consumption on a day with an average temperature of \(37\,^{\circ}\)C. Comment on the reliability of your estimate.
- (e) Rewrite the equation of your regression line found in part (c) in terms of \(y\) and \(F\), where \(F\) is the average daily temperature in degrees Fahrenheit.
Show full worked solution▾
(a) The scatter diagram of \(y\) against \(x\):
(b)(i) From GC, \(r = 0.89056 \approx 0.891\).
(b)(ii) From GC, \(r = 0.92229 \approx 0.922\).
(c) From the scatter diagram, it is observed that as \(x\) increases, \(y\) increases by increasing amounts, and the product moment correlation coefficient between \(x^2\) and \(y\) is \(0.922\), which is closer to \(1\) than that between \(x\) and \(y\), which is \(0.891\). Hence \(y = a + bx^2\) is the better model.
From GC: \(a = 4.8291 \approx 4.83\) and \(b = 0.014074 \approx 0.0141\). \[ y = 4.83 + 0.0141\,x^2 \]
(d) When \(x = 37\): \[ y = 4.8291 + 0.014074\,(37)^2 \approx 24.1\ \text{MWh} \] Since \(x = 37\) lies within the given range \(20 \le x \le 38\) and \(r \approx 0.922\) is close to \(1\), the estimated electricity consumption is reliable.
(e) \(F = \dfrac{9}{5}C + 32 \Rightarrow C = \dfrac{5}{9}(F - 32)\).
Replacing \(x\) by \(\dfrac{5}{9}(F - 32)\): \[\begin{aligned} y &= 4.8291 + 0.014074\!\left[\dfrac{5}{9}(F - 32)\right]^{\!2}\\ &= 4.8291 + 0.0043438\,(F - 32)^2\\ &\approx 4.83 + 0.00434\,(F - 32)^2 \end{aligned}\]