| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | CI from raw data list |
| Difficulty | Moderate -0.3 This is a straightforward S3 confidence interval question requiring standard procedures: calculating sample mean and unbiased variance (bookwork formulas), finding a symmetric interval using known σ, and constructing a confidence interval with known σ. All techniques are routine applications of formulas with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = \frac{500}{10} = 50\) | M1A1 | |
| \(s_t = \frac{1}{9}\left(25001.74 - \frac{500^2}{10}\right) = 0.193\) | awrt0.193 M1A1 A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| limits are \(S = \pm 1.9465\) | M1 E | |
| \(= (49.08, 50.98)\) | awrt 49(0), 51(0) M1A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Confidence interval is | B1 2.5798 | M1 B1 2.5798 |
| \(\left(50 - \frac{2.5758 \times 0.5}{\sqrt{10}}, 50 + \frac{2.5758 \times 0.5}{\sqrt{10}}\right)\) | M1 B1 2.5798 | |
| \(= (49.59273, 50.4L = 227...)\) | awrt awrt 49.6, 50.4 A1 A1 | (5) |
| use 4 or h-table in (v) instead. | TOTAL 14 |
## Part (a)
$\bar{x} = \frac{500}{10} = 50$ | M1A1 |
$s_t = \frac{1}{9}\left(25001.74 - \frac{500^2}{10}\right) = 0.193$ | awrt0.193 M1A1 A1 | (5)
## Part (b)
limits are $S = \pm 1.9465$ | M1 E |
$= (49.08, 50.98)$ | awrt 49(0), 51(0) M1A1 | (4)
## Part (c)
Confidence interval is | B1 2.5798 | M1 B1 2.5798
$\left(50 - \frac{2.5758 \times 0.5}{\sqrt{10}}, 50 + \frac{2.5758 \times 0.5}{\sqrt{10}}\right)$ | M1 B1 2.5798 |
$= (49.59273, 50.4L = 227...)$ | awrt awrt 49.6, 50.4 A1 A1 | (5)
use 4 or h-table in (v) instead. | | TOTAL 14
---A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg, and gave the following results
49.7, 50.3, 51.0, 49.5, 49.9
50.1, 50.2, 50.0, 49.6, 49.7.
\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates of the mean and variance of the weights of the population of metal containers. [5]
\end{enumerate}
The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Estimate the limits between which 95\% of the weights of metal containers lie. [4]
\item Determine the 99\% confidence interval for the mean weight of metal containers. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2006 Q7 [14]}}