Moderate -0.8 This is a routine S1 statistics question requiring standard application of regression formulas with calculator use. Part (a)(i) involves calculating means and using the regression line formula (typically done with calculator statistics mode), part (ii) is direct substitution, and parts (iii)-(iv) test basic understanding of regression limitations. The calculations are straightforward with no conceptual challenges beyond standard A-level statistics content.
During a particular summer holiday, Rick worked in a fish and chip shop at a seaside resort.
He suspected that the shop's takings, \(\pounds y\), on a weekday were dependent upon the forecast of that day's maximum temperature, \(x ^ { \circ } \mathrm { C }\), in the resort, made at 6.00 pm on the previous day.
To investigate this suspicion, he recorded values of \(x\) and \(y\) for a random sample of 7 weekdays during July.
\(\boldsymbol { x }\)
23
18
27
19
25
20
22
\(\boldsymbol { y }\)
4290
3188
5106
3829
5057
4264
4485
Calculate the equation of the least squares regression line of \(y\) on \(x\).
Estimate the shop's takings on a weekday during July when the maximum temperature was forecast to be \(24 ^ { \circ } \mathrm { C }\).
Explain why your equation may not be suitable for estimating the shop's takings on a weekday during February.
Describe, in the context of this question, a variable other than the maximum temperature, \(x\), that may affect \(y\).
Seren, who also worked in the fish and chip shop, investigated the possible linear relationship between the shop's takings, \(\pounds z\), recorded in \(\pounds 000\) s, and each of two other explanatory variables, \(v\) and \(w\).
She calculated correctly that the regression line of \(z\) on \(v\) had a \(z\)-intercept of - 1 and a gradient of 0.15 .
Draw this line, for values of \(v\) from 0 to 40, on Figure 1 on page 4.
She also calculated correctly that the regression line of \(z\) on \(w\) had a \(z\)-intercept of 5 and a gradient of - 0.40 .
Draw this line, for values of \(w\) from 0 to 10, on Figure 2 below.
\begin{figure}[h]
3
\begin{enumerate}[label=(\alph*)]
\item During a particular summer holiday, Rick worked in a fish and chip shop at a seaside resort.
He suspected that the shop's takings, $\pounds y$, on a weekday were dependent upon the forecast of that day's maximum temperature, $x ^ { \circ } \mathrm { C }$, in the resort, made at 6.00 pm on the previous day.
To investigate this suspicion, he recorded values of $x$ and $y$ for a random sample of 7 weekdays during July.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 23 & 18 & 27 & 19 & 25 & 20 & 22 \\
\hline
$\boldsymbol { y }$ & 4290 & 3188 & 5106 & 3829 & 5057 & 4264 & 4485 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$.
\item Estimate the shop's takings on a weekday during July when the maximum temperature was forecast to be $24 ^ { \circ } \mathrm { C }$.
\item Explain why your equation may not be suitable for estimating the shop's takings on a weekday during February.
\item Describe, in the context of this question, a variable other than the maximum temperature, $x$, that may affect $y$.
\end{enumerate}\item Seren, who also worked in the fish and chip shop, investigated the possible linear relationship between the shop's takings, $\pounds z$, recorded in $\pounds 000$ s, and each of two other explanatory variables, $v$ and $w$.
\begin{enumerate}[label=(\roman*)]
\item She calculated correctly that the regression line of $z$ on $v$ had a $z$-intercept of - 1 and a gradient of 0.15 .
Draw this line, for values of $v$ from 0 to 40, on Figure 1 on page 4.
\item She also calculated correctly that the regression line of $z$ on $w$ had a $z$-intercept of 5 and a gradient of - 0.40 .
Draw this line, for values of $w$ from 0 to 10, on Figure 2 below.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{767ec629-6350-41d9-bbb9-e059a5fd8c70-4_792_604_680_717}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{767ec629-6350-41d9-bbb9-e059a5fd8c70-4_792_696_1692_687}
\end{center}
\end{figure}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2011 Q3 [15]}}