| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Symmetric properties of normal |
| Difficulty | Challenging +1.2 This question requires identifying parameters from graphical properties (symmetry gives μ=35, inflection point gives σ=4), calculating a single probability P(30<X<40), then applying binomial distribution with normal approximation or exact calculation. While it involves multiple steps and two distributions, each step follows standard procedures with no novel insight required, making it moderately above average difficulty. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.05d Sample mean as random variable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mu = 35\) soi | B1 | by symmetry |
| \(\sigma = 4\) seen | B1 | \(35 - 31 = 4\); may be embedded in \(N(35,4^2)\) or \(N(35,16)\); M0 if continuity corrections used |
| Use of \(N(35, 4^2)\) to obtain a value for \(P(30 < X < 40)\) | M1 | |
| \([\text{cdfNormal}(30, 40, 35, 4) =]\ 0.788700\ldots\) BC to 2 or more dp | A1 | |
| Use of \(Y \sim B(50,\ \text{their}\ 0.7887)\) to find \(P(Y \leq K)\) or \(P(Y \geq K)\) | M1 | eg \(\text{cdfBinomial}(50, 0.7887, 45, 50)\); their 0.7887 must be from use of \(N(35, 4^2)\) |
| \([1 - P(Y \leq 44)] = 0.032\) to \(0.034\) BC | A1 |
## Question 15:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mu = 35$ soi | B1 | by symmetry |
| $\sigma = 4$ seen | B1 | $35 - 31 = 4$; may be embedded in $N(35,4^2)$ or $N(35,16)$; **M0** if continuity corrections used |
| Use of $N(35, 4^2)$ to obtain a value for $P(30 < X < 40)$ | M1 | |
| $[\text{cdfNormal}(30, 40, 35, 4) =]\ 0.788700\ldots$ BC to 2 or more dp | A1 | |
| Use of $Y \sim B(50,\ \text{their}\ 0.7887)$ to find $P(Y \leq K)$ or $P(Y \geq K)$ | M1 | eg $\text{cdfBinomial}(50, 0.7887, 45, 50)$; their 0.7887 must be from use of $N(35, 4^2)$ |
| $[1 - P(Y \leq 44)] = 0.032$ to $0.034$ BC | A1 | |
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Please share the actual mark scheme pages (the pages containing the question answers and mark allocations) and I will format them as requested.15 You must show detailed reasoning in this question.
The screenshot in Fig. 15 shows the probability distribution for the continuous random variable $X$, where $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_387_954_1599_260}
\captionsetup{labelformat=empty}
\caption{Fig. 15}
\end{center}
\end{figure}
The distribution is symmetrical about the line $x = 35$ and there is a point of inflection at $x = 31$.\\
Fifty independent readings of $X$ are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .
\hfill \mbox{\textit{OCR MEI Paper 2 2019 Q15 [6]}}