| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2020 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for confidence intervals |
| Difficulty | Standard +0.3 Part (a) is a standard confidence interval calculation using the CLT with large n=600. Part (b) tests understanding of when CLT applies versus using t-distribution, but since the population is stated as normal, the answer is that it wouldn't change conceptually (though would use t-distribution). This is slightly above average due to the conceptual justification required in (b), but remains a fairly routine Further Maths Statistics question. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = 1.88\) | B1 | AWRT 1.88; PI by correct upper or lower limit of CI |
| \(\bar{x} \pm z\sqrt{\frac{s^2}{n}} = 1196 \pm 1.88\frac{98}{\sqrt{600}}\) | M1 | Uses formula for upper or lower limit of CI using their \(z\)-value and sample mean and variance; condone use of \(\sqrt{98}\); may use \(t\)-value |
| \(= (1188.5, 1203.5)\) | A1 | AWRT 1dp |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Yes — sample size has changed and is part of the calculation; a \(t\) distribution would be used instead of \(z\); CI will be wider | E1 | Must state yes and explain that sample size has changed and is part of the calculation |
## Question 3:
### Part 3(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = 1.88$ | B1 | AWRT 1.88; PI by correct upper or lower limit of CI |
| $\bar{x} \pm z\sqrt{\frac{s^2}{n}} = 1196 \pm 1.88\frac{98}{\sqrt{600}}$ | M1 | Uses formula for upper or lower limit of CI using their $z$-value and sample mean and variance; condone use of $\sqrt{98}$; may use $t$-value |
| $= (1188.5, 1203.5)$ | A1 | AWRT 1dp |
### Part 3(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Yes — sample size has changed and is part of the calculation; a $t$ distribution would be used instead of $z$; CI will be wider | E1 | Must state yes and explain that sample size has changed and is part of the calculation |
---3 The mass of male giraffes is assumed to have a normal distribution.
Duncan takes a random sample of 600 male giraffes.\\
The mean mass of the sample is 1196 kilograms.\\
The standard deviation of the sample is 98 kilograms.\\
3
\begin{enumerate}[label=(\alph*)]
\item Construct a 94\% confidence interval for the mean mass of male giraffes, giving your values to one decimal place.\\
3
\item Explain whether or not your answer to part (a) would change if a sample of size 5 was taken with the same mean and standard deviation.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2020 Q3 [4]}}