| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Known variance (z-distribution) |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for estimating a mean and constructing a confidence interval with known variance. It requires only calculating a sample mean and applying the z-interval formula—both routine procedures with no problem-solving or conceptual challenges beyond basic recall. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Est}\,\mu = \text{sample mean} = 5.25\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \((\bar{x}) \pm z\text{SD}\) | M1 | |
| \(\text{SD} = 0.19/\sqrt{5}\) | B1 | With \(\sqrt{5}\) seen |
| \(z = 1.96\) | B1 | |
| \(5.083 < \mu < 5.417\) | A1 4 | Rounding to 5.08, 5.42 |
## Question 1:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Est}\,\mu = \text{sample mean} = 5.25$ | B1 | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $(\bar{x}) \pm z\text{SD}$ | M1 | |
| $\text{SD} = 0.19/\sqrt{5}$ | B1 | With $\sqrt{5}$ seen |
| $z = 1.96$ | B1 | |
| $5.083 < \mu < 5.417$ | A1 **4** | Rounding to 5.08, 5.42 |
---1 A random variable has a normal distribution with unknown mean $\mu$ and known standard deviation 0.19 . In order to estimate $\mu$ a random sample of five observations of the random variable was taken. The values were as follows.
$$\begin{array} { l l l l l }
5.44 & 4.93 & 5.12 & 5.36 & 5.40
\end{array}$$
Using these five values, calculate,\\
(i) an estimate of $\mu$,\\
(ii) a 95\% confidence interval for $\mu$.
\hfill \mbox{\textit{OCR S3 2011 Q1 [5]}}