| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Known variance confidence interval |
| Difficulty | Moderate -0.5 This is a straightforward application of a known variance confidence interval for a normal distribution. Students need to calculate the sample mean, recall the formula for a confidence interval with known variance, and look up the z-value for 90% confidence. It's more routine than average A-level questions since it's a direct application of a standard formula with no conceptual complications, but requires careful arithmetic and knowledge of the normal distribution table. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{x} = 15.37\) | B1 | |
| Standard error \(= \sqrt{\frac{0.9}{10}}\) | B1 | \(SE^2 = \frac{0.9}{10}\) |
| Use of \(\bar{x} \pm z \times SE\) | M1 | FT their \(\bar{x}\) and \(SE \neq \sqrt{0.9}\) |
| \(= 15.37 \pm 1.6449 \times \sqrt{\frac{0.9}{10}}\) | A1 | 1.645 or better |
| \([14.88, 15.86]\) | A1 | cao |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x} = 15.37$ | B1 | |
| Standard error $= \sqrt{\frac{0.9}{10}}$ | B1 | $SE^2 = \frac{0.9}{10}$ |
| Use of $\bar{x} \pm z \times SE$ | M1 | FT their $\bar{x}$ and $SE \neq \sqrt{0.9}$ |
| $= 15.37 \pm 1.6449 \times \sqrt{\frac{0.9}{10}}$ | A1 | 1.645 or better |
| $[14.88, 15.86]$ | A1 | cao |
---\begin{enumerate}
\item Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.
\end{enumerate}
$$\begin{array} { l l l l l l l l l l }
15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1
\end{array}$$
Assuming that these data come from a normal distribution with mean $\mu$ and variance $0 \cdot 9$, calculate a $90 \%$ confidence interval for $\mu$.\\
\hfill \mbox{\textit{WJEC Further Unit 5 2022 Q1 [5]}}