A study into the germination of parsnip seeds in the 1980s produced the following data for the average $C$ number of days, $d$, taken for a seed to germinate at different soil temperatures, $t$, measured in degrees Fahrenheit.

(i) Sketch a scatter diagram of the data. State the product moment correlation coefficient between $d$ and $t$

[2]

(ii) Lim thinks the data can be modelled by the regression equation $d=-a+b u$, where $u=\frac{1}{t}$. Find the values of $a$ and $b$ for Lim’s model, giving the values correct to 3 significant figures. State the product moment correlation coefficient between $d$ and $u$.

[3]

The study also found that, at a soil temperature of 86 degrees Fahrenheit, parsnip seeds took an average of 32 days to germinate.

(iii) Determine whether Lim’s model fits this additional data.

[1]

A temperature of $F$ degrees Fahrenheit is equivalent to a temperature of $C$ degrees Celsius, where $C=\frac{5}{9}(F-32)$

(iv) Write Lim’s equation from part (ii) in terms of $d$ and $T$, where $T$ is the temperature in degrees Celsius.

[2]

(i) $r=-0.848$

(ii) $a=145, b=9460, r=0.941$

(iv) $d =-145.13+\frac{47282}{9 T+160}$

(ii) $d=-a+b\left(\frac{1}{t}\right)$

From GC,

$$

\begin{array}{l}

d=9456.4\left(\frac{1}{t}\right)-145.13 [B1] \\

\therefore \quad b \approx 9460, a \approx 145

\end{array}

$$

[Al]

Product moment correlation coefficient, $r \approx 0.941$

[A1]

(iii) Lim’s model does not fit this additional data of 86 degrees Fahrenheit as this value is outside of the range of Lim’s model, i.e. $32<t<68$. Extrapolation is not recommended.

[Al]

(iv) Given $C=\frac{5}{9}(F-32)$

$$

\Rightarrow F=\frac{9 C}{5}+32

$$

If $T$ is the temperature in degrees Celsius, $t$ is the temperature in degrees Fahrenheit.

$$

t=\frac{9 T}{5}+32$$

[B1]

From (ii),

$$

\begin{aligned}

d &=9456.4\left(\frac{1}{t}\right)-145.13 \\

d &=9456.4\left[\frac{1}{\left(\frac{9 T}{5}+32\right)}\right]-145.13 \\

&=-145.13+\frac{47282}{9 T+160}

\end{aligned}

$$

[A1]