| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Select random sample using random numbers |
| Difficulty | Easy -1.8 This is a straightforward S1/S2 level sampling question requiring only basic recall of sampling methods. Part (a) involves simple reading of a random number table (165, 008), while parts (b)-(d) test standard definitions of systematic, stratified, and quota sampling with no calculation or problem-solving required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.01d Select/critique sampling: in context |
| 319 | 952 | 241 | 343 | 278 | 811 | 394 | 165 | 008 | 413 | 063 | 179 | 749 |
| 722 | 962 | 334 | 461 | 267 | 114 | 806 | 992 | 414 | 837 | 837 | 657 | 339 |
| 470 | 684 | 554 | 127 | 067 | 459 | 142 | 920 | 144 | 575 | 311 | 605 | 412 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 165, 8 | B1 | 165 followed by 8 or 008 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Select every \(6^{\text{th}}\) person | B1 | For selecting every 6th name on the list |
| Selecting a random number between 1 and 6, or selecting a random number and then looping back to start when you reach the end | dB1 | Dependent on 1st B1. For idea of using random numbers to select first from 1 to 6 or 0 to 5, or selecting a random number between 1 and 300 and looping back when end of list reached |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The list is alphabetical and has not been sorted by gender | B1 | Comment implying list/sampling frame has not been sorted by gender. B0 for "the ordered list is not truly random". B0 for "sample does not divide the members into gender." |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Label male members 1–180, female members 1–120 | M1 | For suitable labelling of all 180 males and all 120 females. Allow labelling female members 181–300. Also allow male members 0–179 and female members 0 to 119 or 180 to 299 |
| Use random numbers to select... | M1 | For use of random numbers to select males and females |
| Simple random sample of 30 male members and 20 female members | A1 | For 30 males and 20 females (dependent on 2nd M1 only). Note: a simple random sample of 30 males and 20 females scores 2nd M1 and A1 |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Any one of: A stratified sample is not biased as members are chosen randomly; or you can estimate the sampling errors for a stratified sample; or it gives more accurate estimates as it is a random process; or a quota sample may be biased whereas a stratified sample is not; or it's not possible to estimate/find the sampling errors for a quota sample (whereas you can for a stratified sample) | B1 | B0 for "a stratified sample can reflect the population structure." B0 for "estimates obtained from each of the strata." |
| [1] | ||
| Total: 8 |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| 165, 8 | B1 | 165 followed by 8 or 008 |
| | **[1]** | |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Select every $6^{\text{th}}$ person | B1 | For selecting every 6th name on the list |
| Selecting a random number between 1 and 6, **or** selecting a random number and then looping back to start when you reach the end | dB1 | Dependent on 1st B1. For idea of using random numbers to select first from 1 to 6 or 0 to 5, **or** selecting a random number between 1 and 300 and looping back when end of list reached |
| | **[2]** | |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| The list is alphabetical and has not been **sorted** by gender | B1 | Comment implying list/sampling frame has **not been sorted** by gender. B0 for "the ordered list is not truly random". B0 for "sample does not divide the members into gender." |
| | **[1]** | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Label male members 1–180, female members 1–120 | M1 | For suitable labelling of all 180 males and all 120 females. Allow labelling female members 181–300. Also allow male members 0–179 and female members 0 to 119 or 180 to 299 |
| Use random numbers to select... | M1 | For use of random numbers to select males and females |
| Simple random sample of 30 male members and 20 female members | A1 | For 30 males and 20 females **(dependent on 2nd M1 only)**. Note: a simple random sample of 30 males and 20 females scores 2nd M1 and A1 |
| | **[3]** | |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| **Any one of:** A stratified sample is not biased as members are chosen randomly; **or** you can estimate the sampling errors for a stratified sample; **or** it gives more accurate estimates as it is a random process; **or** a quota sample may be biased whereas a stratified sample is not; **or** it's not possible to estimate/find the sampling errors for a quota sample (whereas you can for a stratified sample) | B1 | B0 for "a stratified sample can reflect the population structure." B0 for "estimates obtained from each of the strata." |
| | **[1]** | |
| | **Total: 8** | |
---\begin{enumerate}
\item A tennis club's committee wishes to select a sample of 50 members to fill in a questionnaire about the club's facilities. The 300 members, of whom 180 are males, are listed in alphabetical order and numbered $1 - 300$ in the club's membership book.
\end{enumerate}
The club's committee decides to use a random number table to obtain its sample.\\
The first three lines of the random number table used are given below.
\begin{center}
\begin{tabular}{ l l l l l l l l l l l l l }
319 & 952 & 241 & 343 & 278 & 811 & 394 & 165 & 008 & 413 & 063 & 179 & 749 \\
722 & 962 & 334 & 461 & 267 & 114 & 806 & 992 & 414 & 837 & 837 & 657 & 339 \\
470 & 684 & 554 & 127 & 067 & 459 & 142 & 920 & 144 & 575 & 311 & 605 & 412 \\
\end{tabular}
\end{center}
Starting with the top left-hand corner (319) and working across, the committee selects 50 random numbers. The first 2 suitable numbers are 241 and 278. Numbers greater than 300 are ignored.\\
(a) Find the next two suitable numbers.
When the club's committee looks at the members corresponding to their random numbers they find that only 1 female has been selected.\\
The committee does not want to be accused of being biased towards males so considers using a systematic sample instead.\\
(b) (i) Explain clearly how the committee could take a systematic sample.\\
(ii) Explain why a systematic sample may not give a sample that represents the proportion of males and females in the club.
The committee decides to use a stratified sample instead.\\
(c) Describe how to choose members for the stratified sample.\\
(d) Explain an advantage of using a stratified sample rather than a quota sample.\\
\hfill \mbox{\textit{Edexcel S3 2014 Q1 [8]}}