| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2023 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Known variance (z-distribution) |
| Difficulty | Standard +0.3 This is a straightforward application of confidence interval formulas for normal distributions with known variance. Part (a) requires calculating sample mean and applying the standard z-interval formula; part (b) tests conceptual understanding of confidence levels; part (c) involves working backwards from a given interval; part (d) requires interpretation. All steps are routine for Further Maths statistics students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\bar{x} = \left(\frac{4014}{90}\right) = 44.6\) | B1 |
| Standard error \(= \sqrt{\frac{4.7^2}{90}}\) | B1 | si |
| Use of \(\bar{x} \pm z \times SE\) | M1 | FT their \(\bar{x}\) and SE provided \(\neq 4.7\) for M1A1. Must show working. |
| \(= 44.6 \pm 2.5758 \times \sqrt{\frac{4.7^2}{90}}\) | A1 | |
| \([43.3, 45.9]\) | A1 | cao. From tables 2.576 |
| (b) | Because the confidence level has decreased, the width is narrower. | E1 |
| (c) | \(\bar{x} = \frac{49.9 + 52.6}{2} = 51\) | |
| Use of \(\bar{x} \pm 1.96 \times \frac{\sigma}{\sqrt{n}}\) | M1 | |
| Upper limit of 95% CI is given by \(51 + 1.96 \times \frac{\sigma}{\sqrt{100}} = 52.6\) oe | A1 | |
| OR | ||
| \(2 \times \frac{\sigma}{\sqrt{100}} \times 1.96 = 3.2\) | (A1) | |
| \(\sigma = 8.163...\) | A1 | cao |
| (d) | Valid comment. E.g. The confidence intervals suggest that athletes who compete in the 400m have lower RHR on average. | E1 |
| Valid reason. E.g. The confidence interval for the athletes who compete in the 400m lies entirely below the confidence interval for the athletes who compete in the discus. | E1 | Condone 'Non-overlapping confidence intervals'. |
| Valid comment and reason. e.g. The RHR of athletes who compete in the discus event are possibly more varied, as the width of the CI is wider and the confidence level is lower. | (E2) | |
| Total [11] |
(a) | $\bar{x} = \left(\frac{4014}{90}\right) = 44.6$ | B1 | si |
| Standard error $= \sqrt{\frac{4.7^2}{90}}$ | B1 | si |
| Use of $\bar{x} \pm z \times SE$ | M1 | FT their $\bar{x}$ and SE provided $\neq 4.7$ for M1A1. Must show working. |
| $= 44.6 \pm 2.5758 \times \sqrt{\frac{4.7^2}{90}}$ | A1 | |
| $[43.3, 45.9]$ | A1 | cao. From tables 2.576 |
(b) | Because the confidence level has decreased, the width is narrower. | E1 | Condone width will be smaller. |
(c) | $\bar{x} = \frac{49.9 + 52.6}{2} = 51$ | | |
| Use of $\bar{x} \pm 1.96 \times \frac{\sigma}{\sqrt{n}}$ | M1 | |
| Upper limit of 95% CI is given by $51 + 1.96 \times \frac{\sigma}{\sqrt{100}} = 52.6$ oe | A1 | |
| OR | | |
| $2 \times \frac{\sigma}{\sqrt{100}} \times 1.96 = 3.2$ | (A1) | |
| $\sigma = 8.163...$ | A1 | cao |
(d) | Valid comment. E.g. The confidence intervals suggest that athletes who compete in the 400m have lower RHR on average. | E1 | FT their CI from (a) |
| Valid reason. E.g. The confidence interval for the athletes who compete in the 400m lies entirely below the confidence interval for the athletes who compete in the discus. | E1 | Condone 'Non-overlapping confidence intervals'. |
| Valid comment and reason. e.g. The RHR of athletes who compete in the discus event are possibly more varied, as the width of the CI is wider and the confidence level is lower. | (E2) | |
| **Total [11]** | | |
---3. Athletes who compete in the 400 m event have resting heart rates (RHR), measured in beats per minute, which are normally distributed with known standard deviation $4 \cdot 7$. A random sample of 90 athletes who compete in the 400 m event is taken. Their resting heart rates are summarised by
$$\sum x = 4014 \quad \text { and } \quad \sum x ^ { 2 } = 182257 .$$
\begin{enumerate}[label=(\alph*)]
\item Find a $99 \%$ confidence interval for the mean of the RHR of athletes who compete in the 400 m event. Give the limits of your interval correct to 1 decimal place.
\item Without doing any further calculation, explain how the width of a $95 \%$ confidence interval would compare to the width of your interval in part (a).
Athletes who compete in the discus event have RHR which are normally distributed with known standard deviation $\sigma$. A random sample of 100 athletes who compete in the discus event is taken. A 95\% confidence interval for the mean of the RHR is calculated as [49•4, 52•6].
\item Determine the value of $\sigma$ that was used to calculate this confidence interval.
\item Referring to the confidence intervals, state, with a reason, what can be said about the RHR of athletes who compete in the 400 m event compared to the RHR of athletes who compete in the discus event.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2023 Q3 [11]}}