| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Moderate -0.3 This is a standard AS-level hypothesis testing question with straightforward binomial probability calculations in part (a) and a routine one-tailed test in part (b). The setup is clear, the hypotheses are obvious (H₀: p=1/3, H₁: p>1/3), and students simply need to apply the standard procedure with no conceptual challenges or novel insights required. Slightly easier than average due to its predictable structure. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a 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/Working | Marks | Guidance |
| Let \(N\) = number of games Naasir wins, \(N\sim B(15,\frac{1}{3})\) | M1 | For selecting binomial model with correct \(n\) and \(p\); award for sight of \(B(15,\frac{1}{3})\) |
| (i) \(P(N=2)=0.059946...\) awrt 0.0599 | A1 | Allow 0.05995 |
| (ii) \(P(N>5)=1-P(N\leqslant5)=0.38162...\) awrt 0.382 | A1 | From calculator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p=\frac{1}{3}\), \(H_1: p>\frac{1}{3}\) | B1 | For correctly stating both hypotheses in terms of \(p\) or \(\pi\); accept \(p=0.\dot{3}\); \(H_1: p\geqslant\frac{1}{3}\) is B0 |
| Let \(X\) = number of games Naasir wins, \(X\sim B(32,\frac{1}{3})\) | M1 | For selecting suitable model; award for sight of \(B(32,\frac{1}{3})\) or implied by 0.03765; can also allow M1 for \(P(X\leqslant15)=0.962\) or better or \(P(X\leqslant14)=0.922\) or better |
| \(P(X\geqslant16)=1-P(X\leqslant15)=0.03765\quad (<0.05)\) | A1 | Sight of \(P(X\geqslant16)\) and answer awrt 0.0377; ALT: CR \(X\geqslant16\), must have seen probabilities: \(P(X\leqslant15)=0.9623\) or \(P(X\leqslant14)=0.9224\) or 0.9223 |
| Significant result so reject \(H_0\); there is evidence to support Naasir's claim | A1 | Conclusion in context mentioning "Naasir" or "his" and "claim" or "method"; or e.g. probability of winning a game is \(>\frac{1}{3}\) or has increased; dependent on M1 and 1st A1 |
## Question 3:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $N$ = number of games Naasir wins, $N\sim B(15,\frac{1}{3})$ | M1 | For selecting binomial model with correct $n$ and $p$; award for sight of $B(15,\frac{1}{3})$ |
| (i) $P(N=2)=0.059946...$ awrt **0.0599** | A1 | Allow 0.05995 |
| (ii) $P(N>5)=1-P(N\leqslant5)=0.38162...$ awrt **0.382** | A1 | From calculator |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p=\frac{1}{3}$, $H_1: p>\frac{1}{3}$ | B1 | For correctly stating both hypotheses in terms of $p$ or $\pi$; accept $p=0.\dot{3}$; $H_1: p\geqslant\frac{1}{3}$ is B0 |
| Let $X$ = number of games Naasir wins, $X\sim B(32,\frac{1}{3})$ | M1 | For selecting suitable model; award for sight of $B(32,\frac{1}{3})$ or implied by 0.03765; can also allow M1 for $P(X\leqslant15)=0.962$ or better or $P(X\leqslant14)=0.922$ or better |
| $P(X\geqslant16)=1-P(X\leqslant15)=0.03765\quad (<0.05)$ | A1 | Sight of $P(X\geqslant16)$ **and** answer awrt 0.0377; ALT: CR $X\geqslant16$, must have seen probabilities: $P(X\leqslant15)=0.9623$ or $P(X\leqslant14)=0.9224$ or 0.9223 |
| Significant result so reject $H_0$; there is evidence to support Naasir's claim | A1 | Conclusion in context mentioning "Naasir" or "his" and "claim" or "method"; or e.g. probability of winning a game is $>\frac{1}{3}$ or has increased; dependent on M1 and 1st A1 |
---\begin{enumerate}
\item Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is $\frac { 1 } { 3 }$\\
Naasir and his friends play the game 15 times.\\
(a) Find the probability that Naasir wins\\
(i) exactly 2 games,\\
(ii) more than 5 games.
\end{enumerate}
Naasir claims he has a method to help him win more than $\frac { 1 } { 3 }$ of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.\\
(b) Stating your hypotheses clearly, test Naasir's claim at the $5 \%$ level of significance.
\hfill \mbox{\textit{Edexcel AS Paper 2 2018 Q3 [7]}}