| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Known variance (z-distribution) |
| Difficulty | Moderate -0.3 This is a straightforward application of standard hypothesis testing procedures with known variance. Part (a) requires calculating a confidence interval using the z-distribution (routine formula application), and part (b) simply asks whether 38 lies within that interval—no additional calculation needed. While it's Further Maths content, the question is entirely procedural with no problem-solving or conceptual challenge beyond basic recall. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{x} = \frac{3540}{100} = 35.4\) | B1 | Correct sample mean |
| \(z = 2.32634787\) | M1 | Correct \(z\) value to at least 3 s.f.; condone \(-2.326\); can be implied by correct CI |
| \(\bar{x} \pm z\sqrt{\frac{\sigma^2}{n}} = 35.4 \pm 2.326\sqrt{\frac{10}{100}}\) | M1 | Uses formula with \(\sqrt{\frac{10}{100}}\) |
| \(= (34.66, 36.14)\) or \(35.4 \pm 0.7357\) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Dante rejects the null hypothesis because 38 is outside the confidence interval. | E1F | States null hypothesis rejected as CI does not contain 38; follow through their CI |
## Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x} = \frac{3540}{100} = 35.4$ | B1 | Correct sample mean |
| $z = 2.32634787$ | M1 | Correct $z$ value to at least 3 s.f.; condone $-2.326$; can be implied by correct CI |
| $\bar{x} \pm z\sqrt{\frac{\sigma^2}{n}} = 35.4 \pm 2.326\sqrt{\frac{10}{100}}$ | M1 | Uses formula with $\sqrt{\frac{10}{100}}$ |
| $= (34.66, 36.14)$ or $35.4 \pm 0.7357$ | A1 | CAO |
## Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Dante rejects the null hypothesis because 38 is outside the confidence interval. | E1F | States null hypothesis rejected as CI does not contain 38; follow through their CI |
---4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes.
4
\begin{enumerate}[label=(\alph*)]
\item A random sample of 100 patients has a total waiting time of 3540 minutes.\\
Calculate a $98 \%$ confidence interval for the population mean of waiting times, giving values to four significant figures.\\
4
\item Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 38 \\
& \mathrm { H } _ { 1 } : \mu \neq 38
\end{aligned}$$
Dante uses a $2 \%$ level of significance.\\
Explain whether Dante accepts or rejects the null hypothesis.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2018 Q4 [5]}}