- A scientist wants to develop a model to describe the relationship between the average daily temperature, \(\mathrm { x } ^ { \circ } \mathrm { C }\), and a household's daily energy consumption, ykWh , in winter.
A random sample of the average temperature and energy consumption are taken from 10 winter days and are summarised below.
$$\begin{gathered}
\sum x = 12 \quad \sum x ^ { 2 } = 24.76 \quad \sum y = 251 \quad \sum y ^ { 2 } = 6341 \quad \sum x y = 284.8
S _ { x x } = 10.36 \quad S _ { y y } = 40.9
\end{gathered}$$
- Find the product moment correlation coefficient between y and x .
- Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\)
- Use your equation to estimate the daily energy consumption when the average daily temperature is \(2 ^ { \circ } \mathrm { C }\)
- Calculate the residual sum of squares (RSS).
The table shows the residual for each value of x .
| \(\mathbf { x }\) | - 0.4 | - 0.2 | 0.3 | 0.8 | 1.1 | 1.4 | 1.8 | 2.1 | 2.5 | 2.6 |
| R esidual | - 0.63 | - 0.32 | - 0.52 | - 0.73 | 0.74 | 2.22 | 1.84 | 0.32 | \(f\) | - 1.88 |
- Find the value of f.
- By considering the signs of the residuals, explain whether or not the linear regression model is a suitable model for these data.