| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Estimate from grouped frequency data |
| Difficulty | Moderate -0.8 This is a straightforward statistics question testing basic sampling terminology, frequency density calculations, and sampling method descriptions. Part (b) requires simple multiplication (frequency density × class width), parts (a) and (c) test recall of standard sampling definitions, and part (d) is incomplete but likely asks about distribution shape. All parts are routine textbook exercises with no problem-solving or novel insight required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.02b Histogram: area represents frequency |
| Age in years at retirement | \(45 -\) | \(50 -\) | \(55 -\) | \(60 -\) | \(65 -\) | \(70 -\) | \(75 - 80\) |
| Frequency density | 0.4 | 1.8 | 2.4 | 2.2 | 1.8 | 1.2 | 0.2 |
| Statistics | |
| n | 500 |
| Mean | 60.0515 |
| \(\sigma\) | 6.5717 |
| s | 6.5783 |
| \(\Sigma x\) | 30025.7601 |
| \(\Sigma \mathrm { x } ^ { 2 }\) | 1824686.322 |
| Min | 36.0793 |
| Q1 | 55.2573 |
| Median | 59.9202 |
| Q3 | 64.4239 |
| Max | 81.742 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Quota sampling | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9\) | B1 | from \(5 \times 1.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Systematic: select every \(24^{\text{th}}\) number on the list | M1 | alternatively select every \(25^{\text{th}}\) number on the list; alternatively select every \(24.8^{\text{th}}\) value on list, rounding as appropriate |
| start randomly between \(n=1\) and \(n \geq 184\) and stop when 200 have been selected (if \(n > 184\), must cycle through list) | A1 | start randomly between \(n=1\) and \(n \geq 25\), and cycle through the list again, stopping when 200 have been selected; start randomly with any value on list, cycle through list repeatedly until 200 items selected |
| Simple random sampling: assign each item in the list a unique number (e.g. from 1 to 4960) | E1 | allow any process where each member of the population has an equal chance of being selected |
| generate random numbers until a sample of 200 has been selected soi | E1 | allow any process where each possible sample has an equal chance of being selected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| as the size of the sample increases, the shape of the distribution appears more and more "Normal" oe | B1 | must refer to shape and closer to Normal shape for larger sample |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| use of \(N(60.0515, 6.5783^2)\) to find \(P(X > 65)\) | M1 | condone use of \(6.5717^2\) or parameters rounded to 3 sf; M0 if continuity correction used or e.g. \(P(X > 64)\) found |
| awrt \(0.23\) | A1 | |
| \(4960 \times \text{their } 0.226\) | M1 | |
| \(1121\) or \(1120\) or \(1119\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. there may be seasonal fluctuations such as teachers retiring in August | B1 | allow any sensible reason in context; do not allow e.g. mean and sd may be different |
## Question 8:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Quota sampling | B1 | |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9$ | B1 | from $5 \times 1.8$ |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Systematic: select every $24^{\text{th}}$ number on the list | M1 | alternatively select every $25^{\text{th}}$ number on the list; alternatively select every $24.8^{\text{th}}$ value on list, rounding as appropriate |
| start randomly between $n=1$ and $n \geq 184$ and stop when 200 have been selected (if $n > 184$, must cycle through list) | A1 | start randomly between $n=1$ and $n \geq 25$, and cycle through the list again, stopping when 200 have been selected; start randomly with any value on list, cycle through list repeatedly until 200 items selected |
| Simple random sampling: assign each item in the list a unique number (e.g. from 1 to 4960) | E1 | allow any process where each member of the population has an equal chance of being selected |
| generate random numbers until a sample of 200 has been selected soi | E1 | allow any process where each possible sample has an equal chance of being selected |
### Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| as the size of the sample increases, the shape of the distribution appears more and more "Normal" oe | B1 | must refer to shape and closer to Normal shape for larger sample |
### Part (e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| use of $N(60.0515, 6.5783^2)$ to find $P(X > 65)$ | M1 | condone use of $6.5717^2$ or parameters rounded to 3 sf; M0 if continuity correction used or e.g. $P(X > 64)$ found |
| awrt $0.23$ | A1 | |
| $4960 \times \text{their } 0.226$ | M1 | |
| $1121$ or $1120$ or $1119$ | A1 | |
### Part (f):
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. there may be seasonal fluctuations such as teachers retiring in August | B1 | allow any sensible reason in context; do not allow e.g. mean and sd may be different |
---8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50 , ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.
\begin{enumerate}[label=(\alph*)]
\item State the name of the sampling method she uses.
Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Age in years at retirement & $45 -$ & $50 -$ & $55 -$ & $60 -$ & $65 -$ & $70 -$ & $75 - 80$ \\
\hline
Frequency density & 0.4 & 1.8 & 2.4 & 2.2 & 1.8 & 1.2 & 0.2 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 8.1}
\end{center}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-08_805_1006_1160_244}
\captionsetup{labelformat=empty}
\caption{Fig. 8.2}
\end{center}
\end{figure}
\item How many people in the sample are aged between 50 and 55?
Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.
\item Describe how Rosella could collect a sample of size 200 from her list using
\begin{itemize}
\item systematic sampling such that every item on the list could be selected,
\item simple random sampling.
\end{itemize}
Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-09_659_1909_388_77}
\captionsetup{labelformat=empty}
\caption{Fig. 8.3}
\end{center}
\end{figure}
\item With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution.
Summary statistics for the sample of 500 are shown in Fig. 8.4.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
\multicolumn{2}{|l|}{Statistics} \\
\hline
n & 500 \\
\hline
Mean & 60.0515 \\
\hline
$\sigma$ & 6.5717 \\
\hline
s & 6.5783 \\
\hline
$\Sigma x$ & 30025.7601 \\
\hline
$\Sigma \mathrm { x } ^ { 2 }$ & 1824686.322 \\
\hline
Min & 36.0793 \\
\hline
Q1 & 55.2573 \\
\hline
Median & 59.9202 \\
\hline
Q3 & 64.4239 \\
\hline
Max & 81.742 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 8.4}
\end{center}
\end{table}
\item Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month.
\item Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2020 Q8 [12]}}