| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Standard +0.3 This is a straightforward application of hypothesis testing for a binomial distribution with standard parts: stating hypotheses, finding critical region using tables, and calculating Type I and Type II errors. All steps follow routine procedures taught in S2 with no novel problem-solving required, though it does require careful work with binomial probabilities and understanding of error types. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: P(0) = \dfrac{1}{10}\); \(H_1: P(0) < \dfrac{1}{10}\) | B1 | Accept \(p\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| For \(B(30, 0.1)\) | M1 | Used not just stated |
| \(P(X=0) = 0.9^{30}\ [= 0.0424]\ [<0.1]\) | M1 | |
| \(P(X=0 \text{ or } 1) = 0.9^{30} + 30 \times 0.9^{29} \times 0.1 = 0.184\ [>0.1]\) | B1 | Accept \(0.184\) or \(0.183\) |
| Rejection region is 0 zeros | A1 | Dependent on M1 M1 and at least one comparison, no errors seen. SC: one unsupported correct answer \(0.0424\)/\(0.184\) (or \(0.183\)) and correct rejection region scores B1; with comparison with \(0.1\) scores B2. Two unsupported correct answers \(0.0424\) and \(0.184\) (or \(0.183\)) and correct rejection region scores B2 or if with one comparison with \(0.1\) scores B3 |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.0424\) | B1 | FT *their* (b) must have a critical region (only follow though Binomial), dependent on answer \(< 0.1\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Bin}\!\left(30,\, \dfrac{1}{40}\right)\) | B1 | SOI |
| \(1 - 0.975^{30}\) | M1 | FT *their* rr and with \(\text{Bin}(30,\, 1/40)\) |
| \(0.532\) (3dp) | A1 | SC Unsupported correct answer scores B2 only |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Not concluding that the probability is less than \(\dfrac{1}{10}\), when in fact it is | B1 | In context |
| 1 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: P(0) = \dfrac{1}{10}$; $H_1: P(0) < \dfrac{1}{10}$ | **B1** | Accept $p$ |
| | **1** | |
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| For $B(30, 0.1)$ | **M1** | Used not just stated |
| $P(X=0) = 0.9^{30}\ [= 0.0424]\ [<0.1]$ | **M1** | |
| $P(X=0 \text{ or } 1) = 0.9^{30} + 30 \times 0.9^{29} \times 0.1 = 0.184\ [>0.1]$ | **B1** | Accept $0.184$ or $0.183$ |
| Rejection region is 0 zeros | **A1** | Dependent on **M1 M1** and at least one comparison, no errors seen. SC: one unsupported correct answer $0.0424$/$0.184$ (or $0.183$) and correct rejection region scores **B1**; with comparison with $0.1$ scores **B2**. Two unsupported correct answers $0.0424$ and $0.184$ (or $0.183$) and correct rejection region scores **B2** or if with one comparison with $0.1$ scores **B3** |
| | **4** | |
---
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.0424$ | **B1** | FT *their* (b) must have a critical region (only follow though Binomial), dependent on answer $< 0.1$ |
| | **1** | |
---
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Bin}\!\left(30,\, \dfrac{1}{40}\right)$ | **B1** | SOI |
| $1 - 0.975^{30}$ | **M1** | FT *their* rr and with $\text{Bin}(30,\, 1/40)$ |
| $0.532$ (3dp) | **A1** | SC Unsupported correct answer scores **B2** only |
| | **3** | |
---
## Question 6(e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Not concluding that the probability is less than $\dfrac{1}{10}$, when in fact it is | **B1** | In context |
| | **1** | |
---6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than $\frac { 1 } { 10 }$. In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the $10 \%$ significance level.
\begin{enumerate}[label=(\alph*)]
\item State suitable null and alternative hypotheses.
\item Find the rejection region for the test.
\item State the probability of a Type I error.\\
It is now given that the machine actually produces a 0 once in every 40 digits, on average.
\item Find the probability of a Type II error.
\item Explain the meaning of a Type II error in this context.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q6 [10]}}