AQA S1 2010 June — Question 6 14 marks

Exam BoardAQA
ModuleS1 (Statistics 1)
Year2010
SessionJune
Marks14
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 least squares regression line from raw data, then recalculation with corrected values. The computations are straightforward (calculating sums, means, Sxx, Sxy) and part (c) provides the summary statistics, reducing computational burden. While multi-part with some interpretation required, it follows a routine textbook template with no conceptual challenges beyond basic formula application.
Spec5.09c Calculate regression line5.09e Use regression: for estimation in context
6 During a study of reaction times, each of a random sample of 12 people, aged between 40 and 80 years, was asked to react as quickly as possible to a stimulus displayed on a computer screen. Their ages, \(x\) years, and reaction times, \(y\) milliseconds, are shown in the table.
PersonAge ( \(\boldsymbol { x }\) years)Reaction time ( \(y \mathrm {~ms}\) )
A41520
B54750
C66650
D72920
E71280
F57620
G60740
H47950
I77970
J65780
K51550
L59730
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
    1. Draw your regression line on the scatter diagram on page 16.
    2. Comment on what this reveals.
  3. It was later discovered that the reaction times for persons E and H had been recorded incorrectly. The values should have been 820 and 590 respectively. After making these corrections, computations gave $$S _ { x x } = 1272 \quad S _ { x y } = 14760 \quad \bar { x } = 60 \quad \bar { y } = 720$$
    1. Using the symbol ⋅ , plot the correct values for persons E and H on the scatter diagram on page 16.
    2. Recalculate the equation of the least squares regression line of \(y\) on \(x\), and draw this regression line on the scatter diagram on page 16.
    3. Hence revise as necessary your comments in part (b)(ii).
      \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-15_2484_1709_223_153}
      \section*{Reaction Times}
      \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-16_1943_1301_351_292}
      \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-17_2484_1707_223_155}
6 During a study of reaction times, each of a random sample of 12 people, aged between 40 and 80 years, was asked to react as quickly as possible to a stimulus displayed on a computer screen.

Their ages, $x$ years, and reaction times, $y$ milliseconds, are shown in the table.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Person & Age ( $\boldsymbol { x }$ years) & Reaction time ( $y \mathrm {~ms}$ ) \\
\hline
A & 41 & 520 \\
\hline
B & 54 & 750 \\
\hline
C & 66 & 650 \\
\hline
D & 72 & 920 \\
\hline
E & 71 & 280 \\
\hline
F & 57 & 620 \\
\hline
G & 60 & 740 \\
\hline
H & 47 & 950 \\
\hline
I & 77 & 970 \\
\hline
J & 65 & 780 \\
\hline
K & 51 & 550 \\
\hline
L & 59 & 730 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$.
\item \begin{enumerate}[label=(\roman*)]
\item Draw your regression line on the scatter diagram on page 16.
\item Comment on what this reveals.
\end{enumerate}\item It was later discovered that the reaction times for persons E and H had been recorded incorrectly. The values should have been 820 and 590 respectively.

After making these corrections, computations gave

$$S _ { x x } = 1272 \quad S _ { x y } = 14760 \quad \bar { x } = 60 \quad \bar { y } = 720$$
\begin{enumerate}[label=(\roman*)]
\item Using the symbol ⋅ , plot the correct values for persons E and H on the scatter diagram on page 16.
\item Recalculate the equation of the least squares regression line of $y$ on $x$, and draw this regression line on the scatter diagram on page 16.
\item Hence revise as necessary your comments in part (b)(ii).

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-15_2484_1709_223_153}
\end{center}

\section*{Reaction Times}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-16_1943_1301_351_292}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-17_2484_1707_223_155}
\end{center}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2010 Q6 [14]}}
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