| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Topic | Linear regression |
| Type | Calculate PMCC from raw data |
| Difficulty | Moderate -0.3 This is a straightforward Further Statistics 1 question testing standard PMCC calculation and regression line concepts. Part (a) requires routine application of the PMCC formula with small dataset (5 points), part (b) tests basic interpretation, part (c) tests understanding of linear transformation properties (standard bookwork), and the second section involves standard regression calculations with summaries provided. All techniques are direct applications of formulas with no novel problem-solving required. Slightly easier than average A-level due to small datasets and provision of summary statistics, though being Further Maths content provides some baseline difficulty. |
| Spec | 2.02c Scatter diagrams and regression lines5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc |
| \(x\) | 7 | 8 | 12 | 6 | 4 |
| \(y\) | 20 | 16 | 7 | 17 | 23 |
| Account | A | B | C | D | E | F | G | H |
| \(p\) | 1.6 | 2.1 | 2.4 | 2.7 | 2.8 | 3.3 | 5.2 | 8.4 |
| \(q\) | 1.6 | 2.3 | 2.2 | 2.2 | 3.1 | 2.9 | 7.6 | 4.8 |
1 Five observations of bivariate data $( x , y )$ are given in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 7 & 8 & 12 & 6 & 4 \\
\hline
$y$ & 20 & 16 & 7 & 17 & 23 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the value of Pearson's product-moment correlation coefficient.
\item State what your answer to part (a) tells you about a scatter diagram representing the data.
\item A new variable $a$ is defined by $a = 3 x + 4$. Dee says "The value of Pearson's product-moment correlation coefficient between $a$ and $y$ will not be the same as the answer to part (a)."
State with a reason whether you agree with Dee.
An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods.
The profit in the first year for each account is $p \%$ and the profit in the second year for each account is $q \%$.
The results are shown in the table and in the scatter diagram.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Account & A & B & C & D & E & F & G & H \\
\hline
$p$ & 1.6 & 2.1 & 2.4 & 2.7 & 2.8 & 3.3 & 5.2 & 8.4 \\
\hline
$q$ & 1.6 & 2.3 & 2.2 & 2.2 & 3.1 & 2.9 & 7.6 & 4.8 \\
\hline
\end{tabular}
\end{center}
$n = 8 \quad \Sigma p = 28.5 \quad \Sigma q = 26.7 \quad \Sigma p ^ { 2 } = 136.35 \quad \Sigma q ^ { 2 } = 116.35 \quad \Sigma p q = 116.70$\\
\includegraphics[max width=\textwidth, alt={}, center]{4c7546b9-03ee-47a1-915f-41e2b4ca19c0-03_762_1248_906_260}\\
(a) State which, if either, of the variables $p$ and $q$ is independent.\\
(b) Calculate the equation of the regression line of $q$ on $p$.\\
(c) (i) Use the regression line to estimate the value of $q$ for an investment account for which $p = 2.5$.\\
(ii) Give two reasons why this estimate could be considered reliable.
\item Comment on the reliability of using the regression line to predict the value of $q$ when $p = 7.0$.
\end{enumerate}
\hfill \mbox{\textit{OCR FS1 AS 2021 Q1 [5]}}