| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Moderate -0.3 This is a straightforward one-tail z-test with all information provided directly (n=100, σ=5, x̄=29, μ₀=30). Part (a) is conceptual reasoning about destructive testing, part (b) is routine hypothesis test mechanics, and part (c) tests understanding of CLT applicability. The large sample size makes calculations simple, and no novel problem-solving is required—just standard procedure application, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Fireworks are destroyed when tested | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): Pop mean time lasted (or \(\mu\)) \(= 30\); \(H_1\): Pop mean time lasted (or \(\mu\)) \(< 30\) | B1 | Not just 'mean' |
| \(\pm\dfrac{29-30}{\dfrac{5}{\sqrt{100}}}\) | M1 | For standardising. Must have \(\sqrt{100}\). Use of totals \(N(3000, 2500)\) giving \(\dfrac{(2900-3000)}{\sqrt{2500}}\) scores M1. No mixed methods |
| \(\pm -2\) | A1 | |
| \(-2 > -2.326\) [Do not reject \(H_0\)] | M1 | Accept \(-2.326\) to \(-2.329\). Valid comparison or area comparison \(0.0228 > 0.01\) or \(0.9772 < 0.99\). Accept CR method \(28.837 < 29\) or \(30.163 > 30\) |
| There is not enough evidence that mean time lasted is less than 30 seconds OR Not enough evidence to support the inspector's suspicion | A1 FT | In context (if used need mean or time / condone average instead of mean), not definite. No contradictions. Note 2 tailed test can score B0 M1 A1 M1 (comparison with \(2.574\)–\(2.579\)) A0 (no FT) |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Yes. Because population distribution is unknown [condone not Normal] | B1 | Both needed. Condone \(X\) for parent population |
| 1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Fireworks are destroyed when tested | **B1** | |
| | **1** | |
---
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: Pop mean time lasted (or $\mu$) $= 30$; $H_1$: Pop mean time lasted (or $\mu$) $< 30$ | **B1** | Not just 'mean' |
| $\pm\dfrac{29-30}{\dfrac{5}{\sqrt{100}}}$ | **M1** | For standardising. Must have $\sqrt{100}$. Use of totals $N(3000, 2500)$ giving $\dfrac{(2900-3000)}{\sqrt{2500}}$ scores **M1**. No mixed methods |
| $\pm -2$ | **A1** | |
| $-2 > -2.326$ [Do not reject $H_0$] | **M1** | Accept $-2.326$ to $-2.329$. Valid comparison or area comparison $0.0228 > 0.01$ or $0.9772 < 0.99$. Accept CR method $28.837 < 29$ or $30.163 > 30$ |
| There is not enough evidence that mean time lasted is less than 30 seconds OR Not enough evidence to support the inspector's suspicion | **A1 FT** | In context (if used need mean or time / condone average instead of mean), not definite. No contradictions. Note 2 tailed test can score **B0 M1 A1 M1** (comparison with $2.574$–$2.579$) **A0** (no FT) |
| | **5** | |
---
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Yes. Because population distribution is unknown [condone not Normal] | **B1** | Both needed. Condone $X$ for parent population |
| | **1** | |
---4 A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last a shorter time than this, on average. He takes a random sample of 100 fireworks of this kind. Each firework in the sample is lit and the time it lasts is noted.
\begin{enumerate}[label=(\alph*)]
\item Give a reason why it is necessary to take a sample rather than testing all the fireworks of this kind.\\
It is given that the population standard deviation of the times that fireworks of this kind last is 5 seconds.
\item The mean time lasted by the 100 fireworks in the sample is found to be 29 seconds.
Test the inspector's suspicion at the $1 \%$ significance level.
\item State with a reason whether the Central Limit theorem was needed in the solution to part (b).
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q4 [7]}}