Problem Statement
An atmospheric scientist records temperature $T$ in $^{\circ}\text{C}$ at different altitudes $h$ in thousands of metres above sea level:
$$\begin{array}{|l|c|c|c|c|c|c|}\hline \text{Altitude } h \text{ (km)} & 1.0 & 2.0 & 3.5 & 5.0 & 6.5 & 8.0 \\ \hline \text{Temperature } T \ (^{\circ}\text{C}) & 18.2 & 11.5 & 2.1 & -7.8 & -17.4 & -27.1 \\ \hline \end{array}$$
(a) Write down Pearson's correlation coefficient $r$. [1 mark]
(b) Write down the equation of the regression line of $T$ on $h$. [2 marks]
(c) Use your equation to estimate the temperature at an altitude of $4.2\text{ km}$. [2 marks]
(d) State whether your estimate in part (c) is reliable, giving a reason. [1 mark]
Verified Solution & Marking Scheme
Part (a): Pearson correlation r
Using GDC linear regression $$(ax + b)$$: $$r = -0.9998 \approx -1.00$$.
Part (b): Regression equation
$$T = -6.47h + 24.6$$ (or $$T = -6.473h + 24.62$$)
Part (c): Predict at h = 4.2
$$T(4.2) = -6.473(4.2) + 24.62 = -27.19 + 24.62 = -2.57^{\circ}\text{C}$$.
Part (d): Reliability
Reliable, because $$h = 4.2$$ lies well within the sampled domain of altitudes ($$1.0 \leq h \leq 8.0$$), representing interpolation, and $$|r| \approx 1.00$$ indicates an extremely strong linear relationship.