2025 VJC P2 Q11

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.

1. Draw a scatter diagram for these values, labelling the axes clearly.[1]

It is thought that the electricity consumption and average daily temperature can be modelled by one of the formulae\[y = a + bx^2, \quad y = c + dx,\]where \(a\), \(b\), \(c\) and \(d\) are constants.

2. Find the value of the product moment correlation coefficient between[2]
   1. \(x\) and \(y\),
   2. \(x^2\) and \(y\).
3. 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.[3]
4. 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.[2]

In some regions, temperature is measured in degrees Fahrenheit instead of degrees Celsius. A temperature of \(C\) degrees Celsius is equivalent to \(F\) degrees Fahrenheit, where \(F = \frac{9}{5}C + 32\).

5. 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.[1]