| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Find minimum sample size |
| Difficulty | Standard +0.3 This is a straightforward application of standard confidence interval formulas for normal distributions with known variance. Part (a) requires rearranging the width formula to find n, parts (b) and (c)(i) involve routine calculations, and (c)(ii) tests standard definition recall. While it requires multiple steps and understanding of the relationship between confidence intervals and hypothesis tests, all techniques are direct applications of formulae without requiring problem-solving insight or novel approaches. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z = 1.8119\) | B1 | Obtains correct \(z\) value; AWRT 1.81; PI by correct value of \(\sqrt{n}\) or \(n\) |
| \(0.02715 = 1.8119 \times \frac{0.3}{\sqrt{n}}\), so \(\sqrt{n} = 1.8119 \times \frac{0.3}{0.02715}\) | M1 | Uses formula for full or half width of confidence interval using their \(z\)-value to obtain equation and attempts to solve, reaching at least \(\sqrt{n}=\) |
| \(n = 401\) | A1 | Obtains correct value of \(n = 400\) or \(401\) depending on accuracy given for \(z\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((10.773, 10.827)\) | B1 | Obtains correct confidence interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The null hypothesis will be rejected as \(10.7\) is outside the confidence interval | E1F | Evaluates the model by comparing proposed population mean with confidence interval from part (b) and concludes null hypothesis will be rejected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| It is accepted that the mean distance travelled by the player in a football match is \(10.7\) kilometres when it is not | E1 | Interprets Type II error in context |
## Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = 1.8119$ | B1 | Obtains correct $z$ value; AWRT 1.81; PI by correct value of $\sqrt{n}$ or $n$ |
| $0.02715 = 1.8119 \times \frac{0.3}{\sqrt{n}}$, so $\sqrt{n} = 1.8119 \times \frac{0.3}{0.02715}$ | M1 | Uses formula for full or half width of confidence interval using their $z$-value to obtain equation and attempts to solve, reaching at least $\sqrt{n}=$ |
| $n = 401$ | A1 | Obtains correct value of $n = 400$ or $401$ depending on accuracy given for $z$ |
## Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(10.773, 10.827)$ | B1 | Obtains correct confidence interval |
## Question 4(c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The null hypothesis will be rejected as $10.7$ is outside the confidence interval | E1F | Evaluates the model by comparing proposed population mean with confidence interval from part (b) and concludes null hypothesis will be rejected |
## Question 4(c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| It is accepted that the mean distance travelled by the player in a football match is $10.7$ kilometres when it is not | E1 | Interprets Type II error in context |
---4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres.
A random sample of $n$ matches is taken.\\
The distance the player runs in this sample of matches has mean 10.8 kilometres.\\
The sample is used to construct a $93 \%$ confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places.
4
\begin{enumerate}[label=(\alph*)]
\item Find the value of $n$\\
4
\item Find the $93 \%$ confidence interval for the mean, giving the limits to three decimal places.\\
4
\item Alison claims that the population mean distance the player runs is 10.7 kilometres.
She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 10.7 \\
& \mathrm { H } _ { 1 } : \mu \neq 10.7
\end{aligned}$$
4 (c) (i) State, with a reason, whether the null hypothesis will be accepted or rejected.
4 (c) (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error.\\
\includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-06_2488_1730_219_141}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2021 Q4 [7]}}