| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Topic | Confidence intervals |
| Type | Comment on claim using CI |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard calculations with known and unknown variance, followed by a simple interpretation. The calculations are routine (z-interval then t-interval), and the comment in part (iii) requires only basic understanding that 1.8 lies outside both intervals. Slightly above average difficulty due to being Further Maths statistics and requiring two different CI methods, but no novel insight or complex reasoning needed. |
| Spec | 5.05d Confidence intervals: using normal distribution |
**Question 3(i)**
$\bar{x} = 1.675$ **B1**
99% confidence limits are $1.675 \pm 2.576 \times \frac{0.1}{\sqrt{6}}$ (ft on wrong mean) **M1A1↓**
99% confidence interval is $(1.57, 1.78)$ (AWRT) **A1** [4]
**Question 3(ii)**
$s_n = 0.09215$ or $s_{n-1} = 0.1009$ **B1**
$\nu = 5 \Rightarrow t_5(0.99) = 4.032$ **B1**
99% confidence limits are $1.675 \pm 4.032 \times \frac{0.1009}{\sqrt{6}}$ or $1.675 \pm 4.032 \times \frac{0.09215}{\sqrt{5}}$ **M1A1**
99% confidence interval is $(1.51, 1.84)$ (AWRT) **A1** [5]
**Question 3(iii)**
Sensible comment referring to the fact that 1.8 is outside the 1st interval but inside 2nd. (ft on their CI's.) **B1↓** [1] **[10]**3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below.
$$\begin{array} { l l l l l l }
1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77
\end{array}$$
(i) Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a $99 \%$ confidence interval for the mean concentration of the chemical present in the river.\\
(ii) If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised $99 \%$ confidence interval for the mean concentration of the chemical present in the river.\\
(iii) If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (i) and (ii).
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2012 Q3 [10]}}