| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Find alpha from CI width |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formulas. Part (a) involves direct calculation of sample mean and unbiased variance from summary statistics (routine bookwork). Part (b) requires working backwards from a confidence interval boundary to find the confidence level, which adds a small problem-solving element but uses standard normal tables and algebraic manipulation. The question is slightly easier than average as it's purely procedural with no conceptual challenges or novel insights required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks |
|---|---|
| 1(a) | 2048 128 |
| Answer | Marks |
|---|---|
| 80 5 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 79 80 80 79 80 | M1 | Substitution into a correct formula. |
| Answer | Marks |
|---|---|
| 395 | A1 |
| Answer | Marks |
|---|---|
| 1(b) | '4.19' |
| Answer | Marks | Guidance |
|---|---|---|
| 80 | M1 | Use of correct equation with their values. |
| z = 1.748 or 1.747 | A1 | Accept 3sf. |
| Answer | Marks | Guidance |
|---|---|---|
| (ɸ(‘1.748’) = 0.960) ‘0.960’ – (1 – ‘0.960’) | M1 | Correct area using their values. |
| α = 92.0 or 91.9 | A1 | Allow 92 . |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 1:
--- 1(a) ---
1(a) | 2048 128
Est μ = 25.6 or or
80 5 | B1
80 52760 2048 2 1 20482
Est σ2 = − or 52760−
79 80 80 79 80 | M1 | Substitution into a correct formula.
Biased 4.14 scores M0.
1656
= 4.19 (3 sf) or
395 | A1
3
--- 1(b) ---
1(b) | '4.19'
‘25.6’ + z = 26.0
80 | M1 | Use of correct equation with their values.
z = 1.748 or 1.747 | A1 | Accept 3sf.
FT Biased z = 1.758.
(ɸ(‘1.748’) = 0.960) ‘0.960’ – (1 – ‘0.960’) | M1 | Correct area using their values.
α = 92.0 or 91.9 | A1 | Allow 92 .
FT Biased 92.1.
A final answer of 0.92 or 0.919 scores A0.
4
Question | Answer | Marks | GuidanceEach of a random sample of 80 adults gave an estimate, $h$ metres, of the height of a particular building. The results were summarised as follows.
$$n = 80 \quad \sum h = 2048 \quad \sum h^2 = 52760$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the population mean and variance. [3]
\item Using this sample, the upper boundary of an $\alpha\%$ confidence interval for the population mean is 26.0.
Find the value of $\alpha$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q1 [7]}}