| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.8 Part (a) is routine confidence interval calculation. Part (b) requires deeper understanding: recognizing that P(CI entirely above μ) = 0.04 (half of the 8% in upper tail), then calculating 0.04³ for independence—this is non-standard and conceptually subtle. Part (c) tests sampling theory understanding. The combination of routine calculation with a genuinely tricky probability interpretation elevates this above average difficulty. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(7.1 \pm z \times \sqrt{\frac{2.6}{75}}\) | M1 | Expression of correct form must be \(z\) (note MR var \(= 2.6^2\) can score M1) seen |
| \(z = 1.751\) | B1 | |
| \(6.77\) to \(7.43\) (3 sfs) | A1 | Must be an interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.04^3\) | M1 | Allow \(0.08^3\) for M1 |
| \(= 0.000064\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. Particular day or time of day | B1 | Allow "Not random" |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $7.1 \pm z \times \sqrt{\frac{2.6}{75}}$ | M1 | Expression of correct form must be $z$ (note MR var $= 2.6^2$ can score M1) seen |
| $z = 1.751$ | B1 | |
| $6.77$ to $7.43$ (3 sfs) | A1 | Must be an interval |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.04^3$ | M1 | Allow $0.08^3$ for M1 |
| $= 0.000064$ | A1 | |
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. Particular day or time of day | B1 | Allow "Not random" |3
\begin{enumerate}[label=(\alph*)]
\item The waiting time at a certain bus stop has variance 2.6 minutes ${ } ^ { 2 }$. For a random sample of 75 people, the mean waiting time was 7.1 minutes. Calculate a $92 \%$ confidence interval for the population mean waiting time.
\item A researcher used 3 random samples to calculate 3 independent $92 \%$ confidence intervals. Find the probability that all 3 of these confidence intervals contain only values that are greater than the actual population mean.
\item Another researcher surveyed the first 75 people who waited at a bus stop on a Monday morning. Give a reason why this sample is unsuitable for use in finding a confidence interval for the mean waiting time.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2017 Q3 [6]}}