| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Binomial parameters from given information |
| Difficulty | Moderate -0.3 This is a straightforward application of standard statistical techniques: calculating mean and standard deviation from grouped data, identifying binomial parameters from context (n=10 boxes, p from mean), computing expected frequencies, and commenting on goodness of fit. All steps are routine A-level statistics procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.06b Fit prescribed distribution: chi-squared test |
| Number of damaged decorations | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Number of boxes | 19 | 35 | 28 | 13 | 5 | 0 |
| Number of damaged decorations | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Observed number of boxes | 19 | 35 | 28 | 13 | 5 | 0 |
| Expected number of boxes |
| Answer | Marks | Guidance |
|---|---|---|
| mean \(= 1.5\) BC | B1 | AO1.1 |
| sd \(= 1.1\), \(1.10\) or awrt \(1.096\) BC | B1 | AO1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = 10\) and \(p = 0.15\) | B1 | AO3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| N | 0 | 1 |
| O | 19 | 35 |
| E | 19.7 | 34.7 |
| Use of \(B(10, p)\) FT their \(p\) to find at least 2 probabilities | M1 | AO3.4 |
| Multiplying their binomial probabilities by 100 | M1 | AO3.4 |
| All correct to 1 dp or better; must add up to 100 | A1 | AO1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Close match between theoretical (or expected) and observed frequencies so model is a good fit | B1 | AO3.5a |
# Question 13:
## Part (a):
mean $= 1.5$ BC | **B1** | AO1.1 |
sd $= 1.1$, $1.10$ or awrt $1.096$ BC | **B1** | AO1.1 |
**[2 marks]**
## Part (b):
$n = 10$ and $p = 0.15$ | **B1** | AO3.3 |
**[1 mark]**
## Part (c):
| N | 0 | 1 | 2 | 3 | 4 | 5+ |
|---|---|---|---|---|---|---|
| O | 19 | 35 | 28 | 13 | 5 | 0 |
| E | 19.7 | 34.7 | 27.6 | 13.0 | 4.0 | 1.0 |
Use of $B(10, p)$ FT their $p$ to find at least 2 probabilities | **M1** | AO3.4 | Must see at least 2 probabilities
Multiplying their binomial probabilities by 100 | **M1** | AO3.4 | Must see at least 2 expected values
All correct to 1 dp or better; must add up to 100 | **A1** | AO1.1 |
**[3 marks]**
## Part (d):
Close match between theoretical (or expected) and observed frequencies so model is a good fit | **B1** | AO3.5a | FT dependent on award of **M1M1** in (c) and all frequencies calculated
**[1 mark]**
---13 At a certain factory Christmas tree decorations are packed in boxes of 10 .
The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged.
His results are displayed in Fig. 13.1.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of damaged decorations & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline
Number of boxes & 19 & 35 & 28 & 13 & 5 & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 13.1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Calculate
\begin{itemize}
\item the mean number of damaged decorations per box,
\item the standard deviation of the number of damaged decorations per box.
\end{itemize}
It is believed that the number of damaged decorations in a box of 10, $X$, may be modelled by a binomial distribution such that $\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )$.
\item State suitable values for $n$ and $p$.
\item Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to $\mathbf { 1 }$ decimal place.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of damaged decorations & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline
Observed number of boxes & 19 & 35 & 28 & 13 & 5 & 0 \\
\hline
Expected number of boxes & & & & & & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 13.2}
\end{center}
\end{table}
\item Explain whether the model is a good fit for these data.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2021 Q13 [7]}}