| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Validity or suitability of sample |
| Difficulty | Moderate -0.8 This is a straightforward S3 confidence interval question requiring only standard recall and basic calculations: identifying stratified sampling, computing a weighted mean, applying the standard CI formula with known variance, and making a simple interpretation. No problem-solving or novel insight required—purely routine textbook application. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.05d Confidence intervals: using normal distribution |
| Type of student | Sample size | Mean number of hours |
| Arts | 12 | 12.6 |
| Science | 12 | 14.1 |
| Mixture | 8 | 10.2 |
| Answer | Marks |
|---|---|
| Stratified sampling | B1 |
| Total: 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Uses naturally occurring (strata) groupings | B1 | |
| e.g. variance of estimator of population mean is usually reduced, individual strata estimates available | either | B1 |
| Total: 2 marks |
| Answer | Marks |
|---|---|
| \(\bar{x} = \frac{(12 \times 12.6) + (12 \times 14.1) + (8 \times 10.2)}{32}\) | M1 A1 |
| \(= 12.56\) | A1 |
| Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Confidence interval is | M1 | |
| \(12.56 \pm 1.96 \times \frac{2.48}{\sqrt{32}}\) | 1.96 | B1 |
| i.e. \(12.56 \pm 0.859276\ldots\) | A1 | |
| i.e. \((11.70, 13.42)\) | accept (11.7, 13.4) | A1 |
| Total: 4 marks |
| Answer | Marks |
|---|---|
| 12 is within the confidence interval; so the time spent by these students is in agreement with the suggestion of the member of staff. | B1; B1 |
| Total: 2 marks |
## Part (a)
| Stratified sampling | B1 |
| **Total: 1 mark** |
## Part (b)
| Uses naturally occurring (strata) groupings | B1 |
| e.g. variance of estimator of population mean is usually reduced, individual strata estimates available | either | B1 |
| **Total: 2 marks** |
## Part (c)
| $\bar{x} = \frac{(12 \times 12.6) + (12 \times 14.1) + (8 \times 10.2)}{32}$ | M1 A1 |
| $= 12.56$ | A1 |
| **Total: 3 marks** |
## Part (d)
| Confidence interval is | M1 |
| $12.56 \pm 1.96 \times \frac{2.48}{\sqrt{32}}$ | 1.96 | B1 |
| i.e. $12.56 \pm 0.859276\ldots$ | A1 |
| i.e. $(11.70, 13.42)$ | accept (11.7, 13.4) | A1 |
| **Total: 4 marks** |
## Part (e)
| 12 is within the confidence interval; so the time spent by these students is in agreement with the suggestion of the member of staff. | B1; B1 |
| **Total: 2 marks** |
---As part of her statistics project, Deepa decided to estimate the amount of time A-level students at her school spend on private study each week. She took a random sample of students from those studying Arts subjects, Science subjects and a mixture of Arts and Science subjects. Each student kept a record of the time they spent on private study during the third week of term.
\begin{enumerate}[label=(\alph*)]
\item Write down the name of the sampling method used by Deepa. [1]
\item Give a reason for using this method and give one advantage this method has over simple random sampling. [2]
\end{enumerate}
The results Deepa obtained are summarised in the table below.
\begin{tabular}{|c|c|c|}
\hline
Type of student & Sample size & Mean number of hours \\
\hline
Arts & 12 & 12.6 \\
Science & 12 & 14.1 \\
Mixture & 8 & 10.2 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that an estimate of the mean time spent on private study by A level students at Deepa's school, based on these 32 students is 12.56, to 2 decimal places. [3]
\end{enumerate}
The standard deviation of the time spent on private study by students at the school was 2.48 hours.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Assuming that the number of hours spent on private study is normally distributed, find a 95% confidence interval for the mean time spent on private study by A level students at Deepa's school. [4]
\end{enumerate}
A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Comment on this suggestion in the light of your interval. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q6 [12]}}