| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | CI from raw data list |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question with known variance requiring standard formula application. Part (a) involves calculating sample mean and applying the z-interval formula, while part (b) requires algebraic manipulation to find minimum n. All techniques are routine for S3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\bar{x} = \frac{10.01 + 9.97 + 9.93 + ...}{8} = 9.9775\) | M1 | |
| \(95\%\) CI \(\bar{x} \pm 1.96 \times \frac{0.08}{\sqrt{8}}\) | B1M1 | |
| \(95\%\) CI for \(\mu\) \((9.92, 10.03)\) | A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 10.00 is within confidence interval so accept that pump may be performing correctly (although sample mean is low). | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Upper limit of CI is \(9.96 + 1.6449 \times \frac{0.08}{\sqrt{n}} < 10.00\) | B1, M1A1ft | |
| \(\frac{1.6449 \times 0.08}{\sqrt{n}} < 0.04\) | ||
| \(\sqrt{n} > \frac{1.6449 \times 0.08}{0.04}\) | M1 | |
| \(n > 10.82....\) therefore minimum \(n = 11\) | A1 cao | (5) |
| (10 marks) |
**7(a)(i)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = \frac{10.01 + 9.97 + 9.93 + ...}{8} = 9.9775$ | M1 | |
| $95\%$ CI $\bar{x} \pm 1.96 \times \frac{0.08}{\sqrt{8}}$ | B1M1 | |
| $95\%$ CI for $\mu$ $(9.92, 10.03)$ | A1 | (4) |
**7(a)(ii)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| 10.00 is within confidence interval so accept that pump may be performing correctly (although sample mean is low). | B1 | (1) |
**7(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Upper limit of CI is $9.96 + 1.6449 \times \frac{0.08}{\sqrt{n}} < 10.00$ | B1, M1A1ft | |
| $\frac{1.6449 \times 0.08}{\sqrt{n}} < 0.04$ | | |
| $\sqrt{n} > \frac{1.6449 \times 0.08}{0.04}$ | M1 | |
| $n > 10.82....$ therefore minimum $n = 11$ | A1 cao | (5) |
| | (10 marks) | |
**Guidance Notes:**
- (a)(i) 1st M1 attempt to find sample mean
- B1 for correct $z$ value
- A1 limits correct to 2 decimal places (or more)
- (b) B1 for correct $z$ value
- 1st M1A1, ft their $z$ value
---7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable $X$, in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. $X$ is known to have distribution $X \sim \mathrm {~N} \left( \mu , 0.08 ^ { 2 } \right)$
\begin{enumerate}[label=(\alph*)]
\item Eight random tests gave the following values of $x$
$$\begin{array} { l l l l l l l l }
10.01 & 9.97 & 9.93 & 9.99 & 9.90 & 9.95 & 10.13 & 9.94
\end{array}$$
\begin{enumerate}[label=(\roman*)]
\item Find a 95\% confidence interval for $\mu$ to 2 decimal places.
\item Use your result to comment on the accuracy of the petrol gauge.
\end{enumerate}\item A sample mean of 9.96 litres was obtained from a random sample of $n$ tests. A $90 \%$ confidence interval for $\mu$ gave an upper limit of less than 10.00 litres. Find the minimum value of $n$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2014 Q7 [10]}}