| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sample size determination |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formulas: part (a) uses width = 2×1.96×σ/√n to find σ, part (b) checks if 26.0 lies in the interval, and part (c) asks for a textbook explanation about increasing sample size. All parts are routine applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z = 1.96\) | B1 (AO1.1b) | AWRT 1.96; PI by correct value of standard deviation |
| \(0.4 = 1.96 \times \frac{s}{\sqrt{150}}\) | M1 (AO3.1a) | Uses formula for full or half width of CI; PI |
| \(s = 2.50\ °C\) | A1 (AO1.1b) | AWRT 2.50; accept 2.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(25.3 + 0.4 = 25.7\) | B1 (AO2.2a) | Deduces correct upper limit of CI |
| \(26 > 25.7\), so CI does not support Oscar's claim | E1F (AO2.2b) | FT their upper limit of the CI |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Oscar can take a larger random sample | B1 (AO3.5c) | Explains how Oscar could refine model to reduce width of CI |
# Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 1.96$ | B1 (AO1.1b) | AWRT 1.96; PI by correct value of standard deviation |
| $0.4 = 1.96 \times \frac{s}{\sqrt{150}}$ | M1 (AO3.1a) | Uses formula for full or half width of CI; PI |
| $s = 2.50\ °C$ | A1 (AO1.1b) | AWRT 2.50; accept 2.5 |
---
# Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $25.3 + 0.4 = 25.7$ | B1 (AO2.2a) | Deduces correct upper limit of CI |
| $26 > 25.7$, so CI does not support Oscar's claim | E1F (AO2.2b) | FT their upper limit of the CI |
---
# Question 4(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Oscar can take a larger random sample | B1 (AO3.5c) | Explains how Oscar could refine model to reduce width of CI |
---4 Oscar is studying the daily maximum temperature in ${ } ^ { \circ } \mathrm { C }$ in a village during the month of June.
He constructs a $95 \%$ confidence interval of width $0.8 ^ { \circ } \mathrm { C }$ using a random sample of 150 days.
He assumes that the daily maximum temperature has a normal distribution.\\
4
\begin{enumerate}[label=(\alph*)]
\item Find the standard deviation of Oscar's sample, giving your answer to three significant figures.\\
4
\item Oscar calculates the mean of his sample to be $25.3 ^ { \circ } \mathrm { C }$\\
He claims that the population mean is $26.0 ^ { \circ } \mathrm { C }$\\
Explain whether or not his confidence interval supports his claim.\\
4
\item Explain how Oscar could reduce the width of his 95\% confidence interval.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2021 Q4 [6]}}