| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State hypotheses with additional parts |
| Difficulty | Moderate -0.8 This is a straightforward hypothesis testing question requiring standard procedures: stating H₀ and H₁ for a proportion test, calculating P(Type I error) = P(X > 3 | p = 0.05) using binomial tables, and identifying which error type is possible given the test outcome. All parts are routine applications of textbook definitions with no problem-solving or novel insight required. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): Proportion \(= 0.05\) | B1 | |
| \(H_1\): Proportion \(> 0.05\) | ||
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1 - (0.95^{25} + 25 \times 0.95^{24} \times 0.05 + {}^{25}C_2 \times 0.95^{23} \times 0.05^2 + {}^{25}C_3 \times 0.95^{22} \times 0.05^3)\) | M1 | |
| Completely correct expression | A1 | |
| \(0.0341\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Type II | B1 | |
| Will conclude proportion not increased | B1 | |
| Total: 2 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: Proportion $= 0.05$ | B1 | |
| $H_1$: Proportion $> 0.05$ | | |
| **Total: 1** | | |
---
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 - (0.95^{25} + 25 \times 0.95^{24} \times 0.05 + {}^{25}C_2 \times 0.95^{23} \times 0.05^2 + {}^{25}C_3 \times 0.95^{22} \times 0.05^3)$ | M1 | |
| Completely correct expression | A1 | |
| $0.0341$ | A1 | |
| **Total: 3** | | |
---
## Question 2(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Type II | B1 | |
| Will conclude proportion not increased | B1 | |
| **Total: 2** | | |2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.
\begin{enumerate}[label=(\alph*)]
\item State suitable null and alternative hypotheses for the test.
\item Find the probability of a Type I error.\\
In fact 2 of the 25 apples were Grade A .
\item Which of the errors, Type I or Type II, is possible? Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q2 [6]}}