A group of climbers collected information about the height above sea level, $h$ metres, and the air temperature, $t^ ext{C}$, at the same time at 8 different points on the same mountain - Edexcel - A-Level Maths Statistics - Question 6 - 2018 - Paper 1

The correlation coefficient $r = -0.985$ is very close to -1. This strong negative correlation indicates a linear relationship between height and air temperature, supporting the use of a regression equation.

Using the formula for the regression coefficients:

$b = \frac{S_{ht}}{S_h}$

With $S_{ht} = -985.00$ and using the calculated $S_h$ value:

$b = \frac{-985.00}{-17501.25} \approx 0.0563$. Thus, rounded to three significant figures, we have $b \approx 0.0563$.

Next, we find $a$:

$a = \overline{t} - b \overline{h}$

Calculating: $\overline{t} = \frac{61}{8}, \overline{h} = \frac{6370}{8}$, and substituting:

$a = 7.625 - 0.0563 \cdot 796.25 \approx -0.0201$

Thus, the regression line is:

$t = -0.0201 + 0.0563h$.

The value of $a \approx -0.0201$ represents the estimated air temperature at sea level. This means that at a height of 0 metres, the air temperature is predicted to be approximately -0.0201 degrees Celsius.

To estimate the drop in temperature, we use the slope $b$ calculated previously:

$\text{Drop in temperature} \approx b \times \Delta h = 0.0563 \times 150 \approx 8.45$

Hence, the estimated drop in temperature during the climb is approximately 8.45 degrees Celsius.