| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Moderate -0.3 This is a straightforward application of standard S2 procedures: calculating sample mean and unbiased variance estimate, then constructing a t-distribution confidence interval. While it requires careful arithmetic and knowledge of the t-distribution, it involves no problem-solving or conceptual challenges—just direct application of learned formulas with given data. Slightly easier than average due to its routine nature, though the calculation burden prevents it from being trivial. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = \frac{15.8}{10} = 1.58\) | B1 | \(\bar{X} \sim \text{N}\!\left(\mu, \frac{\sigma^2}{10}\right)\) |
| \(s^2 = \frac{25.0592}{9} - \frac{10}{9}(1.58)^2 = 0.01057\) | B2 (3 marks) | AWRT \(0.011\); \((s = 0.1028)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1.58 \pm \frac{s}{\sqrt{10}} \times 1.833\) | M1A1 (ft) | \(1.58 \pm 0.0596\); for \(\nu = 9\); for \(t\) |
| \((1.52, 1.64)\) | B1, B1\(\checkmark\), A1\(\checkmark\) (5 marks) | for interval |
## Question 3:
### Part (a)
$\sum x = 15.8$, $\sum x^2 = 25.0592$
$\bar{x} = \frac{15.8}{10} = 1.58$ | B1 | $\bar{X} \sim \text{N}\!\left(\mu, \frac{\sigma^2}{10}\right)$
$s^2 = \frac{25.0592}{9} - \frac{10}{9}(1.58)^2 = 0.01057$ | B2 (3 marks) | AWRT $0.011$; $(s = 0.1028)$
### Part (b)
90% CI for $\mu$:
$1.58 \pm \frac{s}{\sqrt{10}} \times 1.833$ | M1A1 (ft) | $1.58 \pm 0.0596$; for $\nu = 9$; for $t$
$(1.52, 1.64)$ | B1, B1$\checkmark$, A1$\checkmark$ (5 marks) | for interval
---3 The heights, in metres, of a random sample of 10 students attending Higrade School are recorded below.\\
$\begin{array} { l l l l l l l l l } 1.76 & 1.59 & 1.54 & 1.62 & 1.49 & 1.52 & 1.56 & 1.47 & 1.75 \end{array} 1.50$
Assume that the heights of students attending Higrade School are normally distributed.
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates for the mean and variance of the heights of students attending Higrade School.\\
(3 marks)
\item Construct a 90\% confidence interval for the mean height of students attending Higrade School.\\
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2005 Q3 [8]}}