| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Finding binomial parameters from properties |
| Difficulty | Moderate -0.3 This is a straightforward application of standard formulas: calculating mean and variance from summary statistics, then using the binomial relationship Var(X) = np(1-p) to find parameters. All steps are routine AS-level techniques with no conceptual challenges or novel problem-solving required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{X} = 2.1\) | B1 | |
| \(\frac{604 - 100\times 2.1^2}{99}\) oe | M1 | |
| variance \(= 1.64646\ldots\) | A1 | Accept to 3 or 4 sf or as recurring decimal or accept \(\frac{163}{99}\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n = 10\) | B1 | |
| \(p = 0.21\) | B1FT | Ft from their \(2.1\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{binomPdf}(n, p, 2)\) used | M1 | FT their \(n\) and their calculated \(p\); may be implied by \(0.301\ldots\) |
| \(100 \times \text{binomPdf}(10, 0.21, 2)\) | M1 | FT 100 \(\times\) their prob from Binomial distribution |
| \(f_e = 30.1\) | A1 | Accept 30 for final answer |
| [3] |
## Question 12:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{X} = 2.1$ | B1 | |
| $\frac{604 - 100\times 2.1^2}{99}$ **oe** | M1 | |
| variance $= 1.64646\ldots$ | A1 | Accept to 3 or 4 sf or as recurring decimal or accept $\frac{163}{99}$ |
| **[3]** | | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $n = 10$ | B1 | |
| $p = 0.21$ | B1FT | Ft from their $2.1$ |
| **[2]** | | |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{binomPdf}(n, p, 2)$ used | M1 | FT their $n$ and their calculated $p$; may be implied by $0.301\ldots$ |
| $100 \times \text{binomPdf}(10, 0.21, 2)$ | M1 | FT 100 $\times$ their prob from Binomial distribution |
| $f_e = 30.1$ | A1 | Accept 30 for final answer |
| **[3]** | | |
---12 A manufacturer of steel rods checks the length of each rod in randomly selected batches of 10 rods.
100 batches of 10 rods are checked and $x$, the number of rods in each batch which are too long, is recorded.
Summary statistics are as follows.\\
$n = 100$
$$\sum x = 210 \quad \sum x ^ { 2 } = 604$$
\begin{enumerate}[label=(\alph*)]
\item Calculate
\begin{itemize}
\item the mean number of rods in a batch which are too long,
\item the variance of the number of rods in a batch which are too long.
\end{itemize}
Layla decides to use a binomial distribution to model the number of rods which are too long in a batch of 10 .
\item Write down the parameters that Layla should use in her model.
\item Use Layla's model to determine the expected number of batches out of 100 in which there are exactly 2 rods which are too long.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2021 Q12 [8]}}