| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward application of binomial hypothesis testing with standard steps. Part (a) involves basic binomial probability calculations (single calculation for exact probability, simple mean formula). Part (b) is a routine one-tailed test at 10% level with clear setup - students just need to follow the standard procedure of stating hypotheses, finding critical region from tables, comparing observed value, and concluding. The question is slightly easier than average because it provides cumulative probability tables, uses a common significance level, and requires no unusual insights beyond the standard hypothesis testing framework taught at AS level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.24706\) | B1 | AWRT 0.247; accept 0.25 and % |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Mean} = np = 7 \times 0.7 = 4.9\) | B1 | CAO; do not ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Y\) is 'No. of times the coin lands heads'; \(H_0: p = 0.7\); \(H_1: p > 0.7\) | B1 | States both hypotheses correctly for a one-tailed test; accept population proportion for \(p\); accept 70% but not \(H_0 = 0.7\); or \(x=\) or \(\bar{x}=\) or \(\mu=\) |
| Under \(H_0: Y \sim B(35,\ 0.7)\) | M1 | States model used; PI by AWRT 0.13, 0.068, 0.038, 0.065, 0.87; or critical region \(>28\) |
| \(P(Y \geq 28) = 1 - P(Y \leq 27) = 1 - 0.86735\ldots = 0.13265 = 0.133\) | A1 | Evaluates \(P(Y \geq 28)\) using calculator \(= 0.13\) (AWRT) OE |
| As \(0.133 > 0.1\), accept \(H_0\) | A1 | Compares 0.13 to 0.1, or 0.87 to 0.9, or 28 to justified critical region of 29 or more; and makes appropriate inference |
| There is insufficient evidence to suggest that the probability of getting a head is more than 0.7. | R1 | Concludes correctly in context; CSO 'insufficient evidence' OE required; only award for full complete correct solution |
## Question 16(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.24706$ | B1 | AWRT 0.247; accept 0.25 and % |
## Question 16(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Mean} = np = 7 \times 0.7 = 4.9$ | B1 | CAO; do not ISW |
## Question 16(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Y$ is 'No. of times the coin lands heads'; $H_0: p = 0.7$; $H_1: p > 0.7$ | B1 | States both hypotheses correctly for a one-tailed test; accept population proportion for $p$; accept 70% but not $H_0 = 0.7$; or $x=$ or $\bar{x}=$ or $\mu=$ |
| Under $H_0: Y \sim B(35,\ 0.7)$ | M1 | States model used; PI by AWRT 0.13, 0.068, 0.038, 0.065, 0.87; or critical region $>28$ |
| $P(Y \geq 28) = 1 - P(Y \leq 27) = 1 - 0.86735\ldots = 0.13265 = 0.133$ | A1 | Evaluates $P(Y \geq 28)$ using calculator $= 0.13$ (AWRT) OE |
| As $0.133 > 0.1$, accept $H_0$ | A1 | Compares 0.13 to 0.1, or 0.87 to 0.9, or 28 to justified critical region of 29 or more; and makes appropriate inference |
| There is insufficient evidence to suggest that the probability of getting a head is more than 0.7. | R1 | Concludes correctly in context; CSO 'insufficient evidence' OE required; only award for full complete correct solution |16 It is believed that a coin is biased so that the probability of obtaining a head when the coin is tossed is 0.7
16
\begin{enumerate}[label=(\alph*)]
\item Assume that the probability of obtaining a head when the coin is tossed is indeed 0.7\\
16 (a) (i) Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.\\
16 (a) (ii) Find the mean number of heads obtained from 7 tosses of the coin.\\
16
\item Harry believes that the probability of obtaining a head for this coin is actually greater than 0.7
To test this belief he tosses the coin 35 times and obtains 28 heads.
Carry out a hypothesis test at the $10 \%$ significance level to investigate Harry's belief.\\
\includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-24_2492_1721_217_150}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-28_2498_1722_213_147}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2022 Q16 [8]}}