| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sampling method explanation |
| Difficulty | Moderate -0.8 This question tests basic understanding of sampling methods, Type I error definition, and a straightforward hypothesis test using CLT. Part (a) requires simple reasoning about bias, (b) is definitional recall, (c) is a standard one-sample z-test calculation with given values, and (d) asks for a textbook explanation of why CLT is needed. All parts are routine applications with no problem-solving or novel insight required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.05a Hypothesis testing language: null, alternative, p-value, significance5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Later customers might spend times different from first ones | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.02\) | B1 | |
| Concluding that \(\mu \neq 6.0\), when actually \(\mu = 6.0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \mu = 6.0\), \(H_1: \mu \neq 6.0\) | B1 | |
| \(\frac{6.8 - 6.0}{\sqrt{\frac{4.8}{50}}}\) | M1 | |
| \(2.582\) | A1 | |
| comp \(2.326\) | M1 | |
| Evidence that \(\mu \neq 6.0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Population distribution unknown | B1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Later customers might spend times different from first ones | B1 | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.02$ | B1 | |
| Concluding that $\mu \neq 6.0$, when actually $\mu = 6.0$ | B1 | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu = 6.0$, $H_1: \mu \neq 6.0$ | B1 | |
| $\frac{6.8 - 6.0}{\sqrt{\frac{4.8}{50}}}$ | M1 | |
| $2.582$ | A1 | |
| comp $2.326$ | M1 | |
| Evidence that $\mu \neq 6.0$ | A1 | |
---
## Question 7(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Population distribution unknown | B1 | |7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.
\begin{enumerate}[label=(\alph*)]
\item He considers choosing the first 50 customers who visit the information desk.
Explain why this method is unsuitable.\\
The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean $\mu$ and variance 4.8. The researcher knows that in the past the value of $\mu$ was 6.0. He wishes to test, at the $2 \%$ significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
\item State the probability of making a Type I error and explain what is meant by a Type I error in this context.
\item Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
\item Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q7 [9]}}