AQA S1 2012 June — Question 3 11 marks

Exam BoardAQA
ModuleS1 (Statistics 1)
Year2012
SessionJune
Marks11
PaperDownload PDF ↗
TopicLinear regression
TypeCalculate y on x from raw data table
DifficultyModerate -0.3 This is a standard S1 linear regression question requiring calculation of regression line from summary statistics (Sxx, Sxy), plotting lines, and interpreting intersection. The calculations are routine and formulaic, though the 12 data points require careful arithmetic. The comparison in part (c)(iii) requires only basic interpretation of gradients and intercepts, making this slightly easier than a typical A-level question.
Spec2.02c Scatter diagrams and regression lines5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression
3 The table shows the maximum weight, \(y _ { A }\) grams, of Salt \(A\) that will dissolve in 100 grams of water at various temperatures, \(x ^ { \circ } \mathrm { C }\).
\(\boldsymbol { x }\)101520253035404550607080
\(\boldsymbol { y } _ { \boldsymbol { A } }\)203548577792101111121137159182
  1. Calculate the equation of the least squares regression line of \(y _ { A }\) on \(x\).
  2. The data in the above table are plotted on the scatter diagram on page 4. Draw your regression line on this scatter diagram.
  3. For water temperatures in the range \(10 ^ { \circ } \mathrm { C }\) to \(80 ^ { \circ } \mathrm { C }\), the maximum weight, \(y _ { B }\) grams, of Salt \(B\) that will dissolve in 100 grams of water is given by the equation $$y _ { B } = 60.1 + 0.255 x$$
    1. Draw this line on the scatter diagram.
    2. Estimate the water temperature at which the maximum weight of Salt \(A\) that will dissolve in 100 grams of water is the same as that of Salt B.
    3. For Salt \(A\) and Salt \(B\), compare the effects of water temperature on the maximum weight that will dissolve in 100 grams of water. Your answer should identify two distinct differences. \section*{Temperatures and Maximum Weights}
      \includegraphics[max width=\textwidth, alt={}]{91466019-8feb-4292-b616-e8e8667e2e54-4_2023_1682_404_173}
3 The table shows the maximum weight, $y _ { A }$ grams, of Salt $A$ that will dissolve in 100 grams of water at various temperatures, $x ^ { \circ } \mathrm { C }$.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 60 & 70 & 80 \\
\hline
$\boldsymbol { y } _ { \boldsymbol { A } }$ & 20 & 35 & 48 & 57 & 77 & 92 & 101 & 111 & 121 & 137 & 159 & 182 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y _ { A }$ on $x$.
\item The data in the above table are plotted on the scatter diagram on page 4.

Draw your regression line on this scatter diagram.
\item For water temperatures in the range $10 ^ { \circ } \mathrm { C }$ to $80 ^ { \circ } \mathrm { C }$, the maximum weight, $y _ { B }$ grams, of Salt $B$ that will dissolve in 100 grams of water is given by the equation

$$y _ { B } = 60.1 + 0.255 x$$
\begin{enumerate}[label=(\roman*)]
\item Draw this line on the scatter diagram.
\item Estimate the water temperature at which the maximum weight of Salt $A$ that will dissolve in 100 grams of water is the same as that of Salt B.
\item For Salt $A$ and Salt $B$, compare the effects of water temperature on the maximum weight that will dissolve in 100 grams of water. Your answer should identify two distinct differences.

\section*{Temperatures and Maximum Weights}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{91466019-8feb-4292-b616-e8e8667e2e54-4_2023_1682_404_173}
\end{center}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2012 Q3 [11]}}
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