| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Topic | Bivariate data |
| Type | Find missing data values |
| Difficulty | Standard +0.3 This is a straightforward two-part question testing basic statistics concepts. Part (a) requires solving a simple quadratic equation from the sum of products, then recognizing perfect linear correlation. Part (b) is a standard Venn diagram probability problem with conditional probability. Both parts use routine techniques with no novel insight required, making this slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| \(x\) | 1 | \(A\) | \(A + 3\) | 10 |
| \(y\) | 2 | \(A - 1\) | \(A\) | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(A(A - 1) + 4(A + 3) + 50 + 2 = 92\) | M1 | Attempt xy products |
| \(A^2 + A - 20 = 0\) or equiv | A1 | Obtain \(A^2 + A - 20 = 0\) or equiv 3 termed expression. State \(A = 4\) only |
| \(A = 4\) | A1 | |
| (ii) The points exactly lie on a straight line | B1 | The line is \(3y - x = 5\) |
| (b) (i) \(240 - x + x + 100 - x = 250\) | M1 | Valid method seen |
| \(X\) or \(P(A \cap B) = 90\) | A1 | Award if 90 seen in the diagram |
| \(\frac{150}{300}\) | A1 | [9] |
| (ii) \(\frac{90}{100}\) | M1 | Use conditional probability. \(\frac{\text{their}x}{100}\) or \(\frac{100}{300}\) |
| B1 | [9] | Obtain 0.9 or equiv |
(a) (i) $A(A - 1) + 4(A + 3) + 50 + 2 = 92$ | M1 | Attempt xy products
$A^2 + A - 20 = 0$ or equiv | A1 | Obtain $A^2 + A - 20 = 0$ or equiv 3 termed expression. State $A = 4$ only
$A = 4$ | A1 |
(ii) The points exactly lie on a straight line | B1 | The line is $3y - x = 5$
(b) (i) $240 - x + x + 100 - x = 250$ | M1 | Valid method seen
$X$ or $P(A \cap B) = 90$ | A1 | Award if 90 seen in the diagram
$\frac{150}{300}$ | A1 | [9] | State $\frac{150}{300}$ aef
(ii) $\frac{90}{100}$ | M1 | Use conditional probability. $\frac{\text{their}x}{100}$ or $\frac{100}{300}$
| B1 | [9] | Obtain 0.9 or equiv\begin{enumerate}[label=(\alph*)]
\item The table below relates the values of two variables $x$ and $y$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & $A$ & $A + 3$ & 10 \\
\hline
$y$ & 2 & $A - 1$ & $A$ & 5 \\
\hline
\end{tabular}
\end{center}
$A$ is a positive integer and $\sum xy = 92$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of $A$. [3]
\item Explain how you can tell that the product-moment correlation coefficient is 1. [1]
\end{enumerate}
\item A music society has 300 members. 240 like Puccini, 100 like Wagner and 50 like neither.
\begin{enumerate}[label=(\roman*)]
\item Calculate the probability that a member chosen at random likes Puccini but not Wagner. [3]
\item Calculate the probability that a member chosen at random likes Puccini given that this member likes Wagner. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q14 [9]}}