| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 5 |
| Topic | Linear combinations of normal random variables |
| Type | Multiple stage process probability |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal variables and the crucial distinction between independent and dependent variables. Part (a) is straightforward application of variance addition for independent normals, but part (b) requires the insight that using the same performance twice means A and C are perfectly correlated (identical), fundamentally changing the variance calculation. This conceptual leap beyond routine application justifies above-average difficulty. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
| Section | Mean | Standard deviation |
| \(A\) | 264 | 13 |
| \(B\) | 173 | 9 |
| \(C\) | 264 | 13 |
| Answer | Marks |
|---|---|
| M1, A1, A1 | Normal, mean \(\mu_A + \mu_B + \mu_C\); Variance 419; Answer, 0.177 or better, www |
| Answer | Marks |
|---|---|
| M1, A1 | Normal, same mean, \(4\sigma_A^2 + \sigma_B^2\); Answer, art 0.245 |
## (a)
$A + B + C \sim N(701, \ldots 419)$
$P(> 720) = 0.176649$
| M1, A1, A1 | Normal, mean $\mu_A + \mu_B + \mu_C$; Variance 419; Answer, 0.177 or better, www |
## (b)
$2A + B - \sim N(701, 757)$
$P(> 720) = 0.244919$
| M1, A1 | Normal, same mean, $4\sigma_A^2 + \sigma_B^2$; Answer, art 0.245 |
---1 The performance of a piece of music is being recorded. The piece consists of three sections, $A , B$ and $C$. The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
Section & Mean & Standard deviation \\
\hline
$A$ & 264 & 13 \\
\hline
$B$ & 173 & 9 \\
\hline
$C$ & 264 & 13 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Assume first that the times for the three sections are independent. Find the probability that the total length of the performance is greater than 720.0 seconds.
\item In fact sections $A$ and $C$ are musically identical, and the recording is made by using a single performance of section $A$ twice, together with a performance of section $B$. In this case find the probability that the total length of the performance is greater than 720.0 seconds.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q1 [5]}}