2.

At a seaside resort the number $X$ of ice-creams sold and the temperature $Y ^ { \circ } \mathrm { F }$ were recorded on 20 randomly chosen summer days. The data can be summarised as follows.

$$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$, giving your answer in the form $y = a + b x$.
\item Explain the significance for the regression line of the quantity $\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }$.
\item It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as $f ^ { \circ }$ Fahrenheit and $c ^ { \circ }$ Centigrade, the relationship between $f$ and $c$ is $c = \frac { 5 } { 9 } ( f - 32 )$.

Find the equation of the new regression line.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q2 [6]}}