| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with standard parameters (n=40, p=0.2 under H₀). Part (a) requires routine calculation of P(X≤4) and comparison to 5%, while part (b) involves finding the critical value by cumulative probability—both are textbook procedures requiring no novel insight, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: P(\text{red}) = 0.2\) \(H_1: P(\text{red}) < 0.2\) | B1 | Allow \(H_0: p = 0.2\), \(H_1: p < 0.2\) |
| \(P(X \leq 4) = 0.8^{40} + 40 \times 0.8^{39} \times 0.2 + {}^{40}C_2 \times 0.8^{38} \times 0.2^2 + {}^{40}C_3 \times 0.8^{37} \times 0.2^3 + {}^{40}C_4 \times 0.8^{36} \times 0.2^4\) | M1 | For full expression seen. Allow one term omitted, incorrect or extra. |
| \(0.0759\) | A1 | SC 0.0759 without working B1. |
| their \('0.0759' > 0.05\) | M1 | Valid comparison (from binomial probs) of their \(P(X \leq 4)\) with \(0.05\). |
| [Do not reject \(H_0\)]. Not enough evidence that it lands on red fewer times than if it were fair or not enough evidence to suggest that the spinner is biased | A1 FT | FT their 0.0759. In context, not definite, no contradictions. |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X \leqslant 3) = \) `\(0.0759\)` \(- {}^{40}C_4 \times 0.8^{36} \times 0.2^4\) | M1 | OE. Attempted. Must be using \(B(40, 0.2)\). Method could be implied by correct answer here. |
| \(= 0.0285\) or \(0.0284\) | *A1 | |
| Largest value of \(r\) is 3 | DA1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: P(\text{red}) = 0.2$ $H_1: P(\text{red}) < 0.2$ | **B1** | Allow $H_0: p = 0.2$, $H_1: p < 0.2$ |
| $P(X \leq 4) = 0.8^{40} + 40 \times 0.8^{39} \times 0.2 + {}^{40}C_2 \times 0.8^{38} \times 0.2^2 + {}^{40}C_3 \times 0.8^{37} \times 0.2^3 + {}^{40}C_4 \times 0.8^{36} \times 0.2^4$ | **M1** | For full expression seen. Allow one term omitted, incorrect or extra. |
| $0.0759$ | **A1** | SC 0.0759 without working B1. |
| their $'0.0759' > 0.05$ | **M1** | Valid comparison (from binomial probs) of their $P(X \leq 4)$ with $0.05$. |
| [Do not reject $H_0$]. Not enough evidence that it lands on red fewer times than if it were fair or not enough evidence to suggest that the spinner is biased | **A1 FT** | FT their 0.0759. In context, not definite, no contradictions. |
| | **5** | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \leqslant 3) = $ `$0.0759$` $- {}^{40}C_4 \times 0.8^{36} \times 0.2^4$ | M1 | OE. Attempted. Must be using $B(40, 0.2)$. Method could be implied by correct answer here. |
| $= 0.0285$ or $0.0284$ | *A1 | |
| Largest value of $r$ is 3 | DA1 | |
---2 A spinner has five sectors, each printed with a different colour. Susma and Sanjay both wish to test whether the spinner is biased so that it lands on red on fewer spins than it would if it were fair. Susma spins the spinner 40 times. She finds that it lands on red exactly 4 times.
\begin{enumerate}[label=(\alph*)]
\item Use a binomial distribution to carry out the test at the $5 \%$ significance level.\\
Sanjay also spins the spinner 40 times. He finds that it lands on red $r$ times.
\item Use a binomial distribution to find the largest value of $r$ that lies in the rejection region for the test at the 5\% significance level.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q2 [8]}}