| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2003 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | CI from raw data list |
| Difficulty | Moderate -0.8 This is a straightforward confidence interval calculation with known σ, requiring only the standard formula with z-critical value lookup and a simple comparison to test a claim. The data is already ordered, calculation is routine, and the hypothesis test interpretation requires no subtlety—just checking if 30 lies in the interval. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(31 \pm 2.326 \times \frac{3}{\sqrt{20}}\) | \(B1\) | For correct mean |
| \(M1\) | Calculation of correct form: \(x\bar{} \pm z \times \frac{s}{\sqrt{n}}\) (must have \(\sqrt{n}\) in denominator) | |
| \(z = 2.326\) | ||
| \(= (29.4, 32.6)\) | \(B1, A1\) | Correct answer |
| (ii) 30% is inside interval. Accept claim (at 2% level) | \(ftB1^*\) | S.R. Solutions not using (i) score \(B1ft\) only for correct working and conclusion |
| \(ftB1^*dep\) |
**(i)** $31 \pm 2.326 \times \frac{3}{\sqrt{20}}$ | $B1$ | For correct mean
| $M1$ | Calculation of correct form: $x\bar{} \pm z \times \frac{s}{\sqrt{n}}$ (must have $\sqrt{n}$ in denominator)
$z = 2.326$ | |
$= (29.4, 32.6)$ | $B1, A1$ | Correct answer
**(ii)** 30% is inside interval. Accept claim (at 2% level) | $ftB1^*$ | S.R. Solutions not using (i) score $B1ft$ only for correct working and conclusion
| $ftB1^*dep$ |3 A consumer group, interested in the mean fat content of a particular type of sausage, takes a random sample of 20 sausages and sends them away to be analysed. The percentage of fat in each sausage is as follows.
$$\begin{array} { l l l l l l l l l l l l l l l l l l l l }
26 & 27 & 28 & 28 & 28 & 29 & 29 & 30 & 30 & 31 & 32 & 32 & 32 & 33 & 33 & 34 & 34 & 34 & 35 & 35
\end{array}$$
Assume that the percentage of fat is normally distributed with mean $\mu$, and that the standard deviation is known to be 3 .\\
(i) Calculate a 98\% confidence interval for the population mean percentage of fat.\\
(ii) The manufacturer claims that the mean percentage of fat in sausages of this type is 30 . Use your answer to part (i) to determine whether the consumer group should accept this claim.
\hfill \mbox{\textit{CAIE S2 2003 Q3 [6]}}