In a study to determine the relationship between the temperature $T$ (measured in degrees Celsius)

Question

In a study to determine the relationship between the temperature $T$ (measured in degrees Celsius) and the altitude $H$ (height above sea level, measured in metres) for a certain mountain, a team of meteorologists ascended and descended the mountain on a day with clear weather and obtained readings of $T$ and $H$.

The table below shows the $10$ readings obtained:

$H$	$2300$	$2390$	$2626$	$2631$	$2700$	$3126$	$3131$	$3400$	$3450$	$3580$
$T$	$11.80$	$9.26$	$10.29$	$7.99$	$9.52$	$5.20$	$7.26$	$3.72$	$4.43$	$2.31$

On the same axes, sketch a scatter diagram of the data and draw the regression line of $T$ against $H$.[2]
State the product moment correlation coefficient between $T$ and $H$. Comment, in context, on the value obtained.[2]

It was found that $5$ of these $10$ readings were recorded between $2\text{ am}$ and $5.30\text{ am}$ when the team of meteorologists were ascending the mountain, while the other $5$ were recorded between $8\text{ am}$ and $11.30\text{ am}$ when they were descending. The routes used for ascending and descending were the same. Between $5.30\text{ am}$ and $8\text{ am}$, the meteorologists remained on the summit.

For these readings, the regression equation for ascending, corrected to $3$ significance figures, is $T = 23.1 - 0.00574H$. The regression equation for descending is $T = a + bH$.

Find, correct to $3$ significance figures, the values of $a$ and $b$.[2]

One of the meteorologists recorded the temperature near the foot of the mountain as $26$ degrees Celsius.

Determine whether it is suitable to use any of the regression equations above to estimate the altitude of the foot of the mountain.[1]

A temperature of $F$ degrees Fahrenheit is equivalent to a temperature of $T$ degrees Celsius, where $F = \frac{9}{5}T + 32$.

The regression equation for ascending, $T = 23.1 - 0.00574H$ can be written in the form $F + c + dH$, where $c$ and $d$ are constants. Find the values of $c$ and $d$.[2]
Use the equation found in part (e) to estimate the temperature in degrees Fahrenheit when the meteorologists were at an altitude of $2800\text{ m}$ while ascending. Give two reasons why you would expect this estimate to be reliable.
[3]

Accepted Answer

(b) ${r = -0.948}$ (${3}$ s.f.) Since ${r}$ is close to ${-1}$, there is a strong negative linear correlation between ${T}$ and ${H}$. The temperature decreases as the altitude increases. (c) ${a = 26.7}$ (${3}$ s.f.), ${b = -0.0063}$ (${3}$ s.f.) (d) None of the regression lines is suitable. This is because ${26}$ degree Celsius is outside the data range for ${T}$. (e) ${c = 73.58}$, ${d = -0.010332}$ (f) When ${H = 2800}$, ${F = 44.6504}$. The estimate is reliable because the value ${H = 2800}$ is within the data range of ${2390 \leq H \leq 3580}$ and the product moment correlation coefficient ${r = -0.999}$ is close to ${-1}$, indicating a strong negative linear correlation between ${F}$ and ${H}$.

2023 RI P2 Q9

2023 RI P2 Q9