| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Topic | Confidence intervals |
| Type | CI from raw data list |
| Difficulty | Standard +0.8 This requires calculating a two-sample t-confidence interval from raw data, involving computing means, pooled variance, and appropriate degrees of freedom. While the calculations are systematic, the small sample sizes, need for pooled variance assumption, and correct application of t-distribution make this more demanding than standard single-sample CI questions. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Village \(P :\) | 4.8 | 3.5 | 2.9 | 3.7 | 4.2 | 4.6 | 5.1 | 3.3 |
| Village \(Q :\) | 2.4 | 1.9 | 2.3 | 3.1 | 2.7 | 2.9 |
(i) $P$: $\bar{x}_1 = 4.0125$, S.D.$_P = 0.73218\ldots$ or $0.78273\ldots$ (or variance) **B1**
$Q$: $\bar{x}_2 = 2.55$, S.D.$_Q = 0.39895\ldots$ or $0.43703\ldots$ (or variance) **B1**
$\hat{\sigma}^2 = \frac{4.28875 + 0.955}{8 + 6 - 2} = 0.43697$ **M1A1**
$t_{12}(0.95) = 1.782$ **B1**
90% confidence limits are:
$1.4625 \pm (1.782 \times 0.66104\ldots \times \sqrt{8^{-1} + 6^{-1}})$ (ft on $t$ value.) **M1A1**✓
90% confidence interval is $(0.826, 2.10)$ (Accept 0.827 from $t = 1.78$.) **A1** [8]
(ii) Distributions of broadband speeds are normal. **B1**
The populations have a common variance. **B1** [2]4 The broadband speed in village $P$ was measured on 8 randomly selected occasions and the broadband speed in village $Q$ was measured on 6 randomly selected occasions. The results, measured in megabits per second, are shown below.
\begin{center}
\begin{tabular}{ l l l l l l l l l }
Village $P :$ & 4.8 & 3.5 & 2.9 & 3.7 & 4.2 & 4.6 & 5.1 & 3.3 \\
Village $Q :$ & 2.4 & 1.9 & 2.3 & 3.1 & 2.7 & 2.9 & & \\
\end{tabular}
\end{center}
(i) Calculate a $90 \%$ confidence interval for the difference in mean broadband speed in these two villages.\\
(ii) State two assumptions that you have made in carrying out the calculation.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q4 [10]}}