| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Topic | Confidence intervals |
| Type | Comment on claim using CI |
| Difficulty | Standard +0.3 This is a standard two-sample t-test confidence interval question with pooled variance. Part (i) requires routine application of a formula (pooled variance, standard error, t-critical value) with straightforward arithmetic. Part (ii) is a simple interpretation of whether zero lies in the interval. While it's a Further Maths topic, the execution is mechanical with no novel insight required, making it slightly easier than average A-level difficulty. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| School \(A\) | \(n _ { A } = 15\) | \(\bar { x } = 53\) | \(\Sigma ( x - \bar { x } ) ^ { 2 } = 925\) |
| School \(B\) | \(n _ { B } = 12\) | \(\bar { y } = 47\) | \(\Sigma ( y - \bar { y } ) ^ { 2 } = 850\) |
**(i)** Assuming common variance is $\sigma^2$
$s^2 = \frac{925 + 850}{15 + 12 - 2} = \frac{1775}{25} = 71$ — M1A1
$(\nu = 25)\ t = 2.485$ — B1
98% confidence limits are $6 \pm 2.485\sqrt{71}\sqrt{\frac{1}{15} + \frac{1}{12}}$ — M1A1$\checkmark$
98% confidence interval is $(-2.11,\ 14.1)$ AWRT — A1 [6]
**(ii)** $0 \in \text{CI} \Rightarrow$ there is insufficient evidence to demonstrate a difference in ability on this test. — M1A1$\checkmark$ [2]
**[Total: 8]**8 Two groups of Year 12 pupils, one at each of schools $A$ and $B$, are given the same mathematics test. The scores, $x$ and $y$, of pupils at schools $A$ and $B$ respectively are summarised as follows.
\begin{center}
\begin{tabular}{ l l l l }
School $A$ & $n _ { A } = 15$ & $\bar { x } = 53$ & $\Sigma ( x - \bar { x } ) ^ { 2 } = 925$ \\
School $B$ & $n _ { B } = 12$ & $\bar { y } = 47$ & $\Sigma ( y - \bar { y } ) ^ { 2 } = 850$ \\
\end{tabular}
\end{center}
(i) Assuming that the two groups are random samples from independent normal populations with means $\mu _ { A }$ and $\mu _ { B }$ respectively and a common, but unknown, variance, construct a $98 \%$ confidence interval for $\mu _ { A } - \mu _ { B }$.\\
(ii) Comment, with a reason, on any difference in ability between the two schools.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q8 [8]}}