| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard procedures: calculating sample statistics, performing a two-tail z-test at a given significance level, and stating when CLT is needed. All steps are routine S2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{x} = 1700/50 = 34\) | B1 | |
| \(\text{Est}(\sigma^2) = \frac{50}{49}\left(\frac{59050}{50} - 34^2\right)\) or \(\frac{1}{49}\left(59050 - \frac{1700^2}{50}\right)\) | M1 | \(\text{Est}(\sigma^2) = \frac{59050}{50} - 34^2\) biased scores M0. |
| \(= 25.5\) (3 sf) or \(\frac{1250}{49}\) | A1 | \(= 25\) scores A0. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): Population mean time \(= 32.4\) \(H_1\): Population mean time \(\neq 32.4\) | B1 | Not just 'mean' but allow just '\(\mu\)'. |
| \(\frac{34 - 32.4}{\frac{\sqrt{25.5}}{\sqrt{50}}}\) | M1 | Must have \(\sqrt{50}\) and not 50. FT *their* mean and var. Can be implied. |
| \(= 2.24\) (3 sf) | A1 | or \(P(\bar{T} > 34) = 0.0125\). SC use of biased var (25) \(z = 2.26\) or \(p = 0.0119\), allow M1A1. |
| \(2.24 < 2.326\) | M1 | Or \(0.0125 > 0.01\) for a valid comparison. |
| [Not reject \(H_0\)] Insufficient evidence that (mean) time has changed | A1FT | In context, not definite. No contradictions. Note: accept CV method \(x_{\text{cri}} = 34.06\) for M1A1. Compares \(34 < 34.06\) for M1, conclusion for A1. Condone \(x = 32.34\) M1A1: compares \(32.4 > 32.34\) for M1, conclusion for A1. |
| 5 | SC for using a one-tail method. Award max 3/5 (B0 M1 A1 M1 A0). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Distribution of times in the population is normal | B1 | Accept answers with no context here. Accept underlying distribution for population. |
| 1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{x} = 1700/50 = 34$ | B1 | |
| $\text{Est}(\sigma^2) = \frac{50}{49}\left(\frac{59050}{50} - 34^2\right)$ or $\frac{1}{49}\left(59050 - \frac{1700^2}{50}\right)$ | M1 | $\text{Est}(\sigma^2) = \frac{59050}{50} - 34^2$ biased scores M0. |
| $= 25.5$ (3 sf) or $\frac{1250}{49}$ | A1 | $= 25$ scores A0. |
| | **3** | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: Population mean time $= 32.4$ $H_1$: Population mean time $\neq 32.4$ | B1 | Not just 'mean' but allow just '$\mu$'. |
| $\frac{34 - 32.4}{\frac{\sqrt{25.5}}{\sqrt{50}}}$ | M1 | Must have $\sqrt{50}$ and not 50. FT *their* mean and var. Can be implied. |
| $= 2.24$ (3 sf) | A1 | or $P(\bar{T} > 34) = 0.0125$. SC use of biased var (25) $z = 2.26$ or $p = 0.0119$, allow M1A1. |
| $2.24 < 2.326$ | M1 | Or $0.0125 > 0.01$ for a valid comparison. |
| [Not reject $H_0$] Insufficient evidence that (mean) time has changed | A1FT | In context, not definite. No contradictions. Note: accept CV method $x_{\text{cri}} = 34.06$ for M1A1. Compares $34 < 34.06$ for M1, conclusion for A1. Condone $x = 32.34$ M1A1: compares $32.4 > 32.34$ for M1, conclusion for A1. |
| | **5** | SC for using a one-tail method. Award max 3/5 (B0 M1 A1 M1 A0). |
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Distribution of times in the population is normal | B1 | Accept answers with no context here. Accept underlying distribution for population. |
| | **1** | |5 Last year the mean time for pizza deliveries from Pete's Pizza Pit was 32.4 minutes. This year the time, $t$ minutes, for pizza deliveries from Pete's Pizza Pit was recorded for a random sample of 50 deliveries. The results were as follows.
$$n = 50 \quad \Sigma t = 1700 \quad \Sigma t ^ { 2 } = 59050$$
\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates of the population mean and variance.
\item Test, at the $2 \%$ significance level, whether the mean delivery time has changed since last year.
\item Under what circumstances would it not be necessary to use the Central Limit Theorem in answering (b)?
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q5 [9]}}