| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard S2 binomial hypothesis testing question requiring finding critical regions and conducting tests. While it involves multiple parts and careful probability calculations, the methods are routine and well-practiced. The two-tailed setup and the need to find probabilities 'as close as possible to 0.005' adds minor complexity, but this is a textbook application of standard procedures without requiring novel insight or problem-solving. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Leave | |
| blank | |
| Q7 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim B(25, 0.40)\) | M1 | Can be implied by any of the correct answers |
| \(P(X \leqslant 3) = 0.0024\) | A1 | Just giving 0.0024 only scores if CR \(X \leqslant 3\) is given; accept \(P(X < 4)\); calc: 0.002366768… |
| \(P(X \geqslant 17) = 0.0043\) | A1 | Just giving 0.0043 only scores if CR \(X \geqslant 17\) is given; accept \(P(X > 16)\); calc: 0.004326388… |
| CR: \(X \leqslant 3\), \(X \geqslant 17\) (o.e.) | A1, A1 | Apply ISW for e.g. \(3 \geqslant X \geqslant 17\) or \(X \leqslant 3\) and \(X \geqslant 17\) etc. If only answer is \(3 \geqslant X \geqslant 17\) award 3rd A1, 4th A0. Accept \(X \leqslant 3 \cap X \geqslant 17\) or \(X \leqslant 3\) and \(X \geqslant 17\) etc. Do not accept probability statements as critical regions. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.0067\) | B1ft | ft for 0.0067 or ft sum of their 2 probabilities (1st A and 2nd A in (a)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.4 \quad H_1: p < 0.4\) | B1 | For both hypotheses in terms of \(p\) or \(\pi\) |
| \([R \sim B(50, 0.4)]\) | — | — |
| \(P(R \leqslant 8) = 0.0002305\ldots\) awrt 0.0002 | M1 | For \(P(R \leqslant 8) =\) awrt 0.0002 or stating CR: \(R \leqslant 11\). Condone writing \(P(X = 8) =\) awrt 0.0002 but award 2nd A0 cso |
| Reject \(H_0\) | A1 | For a correct non-contextual conclusion |
| Evidence that the changes have been successful or there are fewer red sweets (o.e.) | A1cso | For a correct contextual conclusion dependent on all other marks |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(25, 0.40)$ | M1 | Can be implied by any of the correct answers |
| $P(X \leqslant 3) = 0.0024$ | A1 | Just giving 0.0024 only scores if CR $X \leqslant 3$ is given; accept $P(X < 4)$; calc: 0.002366768… |
| $P(X \geqslant 17) = 0.0043$ | A1 | Just giving 0.0043 only scores if CR $X \geqslant 17$ is given; accept $P(X > 16)$; calc: 0.004326388… |
| CR: $X \leqslant 3$, $X \geqslant 17$ (o.e.) | A1, A1 | Apply ISW for e.g. $3 \geqslant X \geqslant 17$ or $X \leqslant 3$ **and** $X \geqslant 17$ etc. If only answer is $3 \geqslant X \geqslant 17$ award 3rd A1, 4th A0. Accept $X \leqslant 3 \cap X \geqslant 17$ or $X \leqslant 3$ and $X \geqslant 17$ etc. Do **not** accept probability statements as critical regions. | **(5)** |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.0067$ | B1ft | ft for 0.0067 or ft sum of their 2 probabilities (1st A and 2nd A in (a)) | **(1)** |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.4 \quad H_1: p < 0.4$ | B1 | For both hypotheses in terms of $p$ or $\pi$ |
| $[R \sim B(50, 0.4)]$ | — | — |
| $P(R \leqslant 8) = 0.0002305\ldots$ awrt 0.0002 | M1 | For $P(R \leqslant 8) =$ awrt 0.0002 **or** stating CR: $R \leqslant 11$. Condone writing $P(X = 8) =$ awrt 0.0002 but award 2nd A0 cso |
| Reject $H_0$ | A1 | For a correct non-contextual conclusion |
| Evidence that the changes have been successful **or** there are fewer red sweets (o.e.) | A1cso | For a correct contextual conclusion dependent on all other marks | **(4)** |
**SC:** Use $B(25, 0.4)$: Can score 1st B1. Also if they get $P(R \leqslant 8) = 0.27353\ldots =$ awrt 0.274 award B1.
**Total: 10 marks**7. A manufacturer produces packets of sweets. Each packet contains 25 sweets. The manufacturer claims that, on average, 40\% of the sweets in each packet are red.
A packet is selected at random.
\begin{enumerate}[label=(\alph*)]
\item Using a $1 \%$ level of significance, find the critical region for a two-tailed test that the proportion of red sweets is 0.40
You should state the probability in each tail, which should be as close as possible to 0.005
\item Find the actual significance level of this test.
The manufacturer changes the production process to try to reduce the number of red sweets. She chooses 2 packets at random and finds that 8 of the sweets are red.
\item Test, at the $1 \%$ level of significance, whether or not there is evidence that the manufacturer's changes to the production process have been successful. State your hypotheses clearly.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Leave \\
blank \\
Q7 \\
\hline
& \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q7 [10]}}