| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Moderate -0.3 This is a standard S2 hypothesis testing question requiring routine application of binomial distribution tables to find critical regions and perform a significance test. Part (a) involves looking up cumulative probabilities to establish a two-tailed critical region, while part (b) is a one-tailed test with clearly stated context. Both parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim B(25, 0.2)\) | M1 | Writing or using \(B(25, 0.2)\) or \(B(25, \frac{1}{5})\) [allow \(Po(5)\)]. May be written in full or implied by a correct CR |
| \(P(X \geq 9) = 0.0468\), \(P(X \leq 1) = 0.0274\) | A1 | Both awrt 0.0468 and awrt 0.0274 seen |
| \(X = [0 \leq]\ X \leq 1\) | A1 | \(X \leq 1\) or \(X < 2\) or \(0 \leq X \leq 1\) or \([0,1]\) or \(0,1\) or equivalent statements. \(X \leq c\) and \(c = 1\) |
| \(9 \leq X\ [\leq 25]\) | A1d | Dependent on seeing a probability from \(B(25, 0.2)\) and \(X \geq 9\) or \(X > 8\) or \(9 \leq X \leq 25\) or list \(9,10,...,25\) or \([9,25]\) or equivalent. \(X \geq c\) and \(c = 9\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: p = 0.2\), \(H_1: p < 0.2\) | B1 | Both hypotheses with \(p\) or \(\pi\), clear which is \(H_0\) and which is \(H_1\) |
| \(P(X \leq 6) = 0.1034\) or \(CR\ X \leq 5\) | M1 | Writing or using \(B(50, 0.2)\) and writing or using \(P(X \leq 6)\) or \(P(X \geq 7)\) on its own. May be implied by a correct CR |
| A1 | awrt 0.103. Allow \(CR\ X \leq 5\) or \(X < 6\). If not using CR, allow awrt 0.897 | |
| Insufficient evidence to reject \(H_0\). Accept \(H_0\), not significant. 6 does not lie in the critical region. | M1d | Dependent on previous M. Correct statement (do not allow contradicting non-contextual statements). ft their Prob/CR compared with \(0.05/6/(0.95\) if using \(0.8979)\). Do not follow through their hypotheses |
| No evidence that increasing the batch size has reduced the percentage of broken pots (oe) or evidence that there is no change in the percentage of broken pots (oe) | A1cso | Conclusion must contain words reduced/no change/not affect oe number/percentage/proportion/probability oe, and pots. All previous marks must be awarded for this mark. Do not allow potters claim/belief is wrong/true. NB Correct contextual statement on its own scores M1A1 |
# Question 1:
## Part (a)
*Note: Allow any letter instead of $X$ or $c$ for this question*
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(25, 0.2)$ | M1 | Writing or using $B(25, 0.2)$ or $B(25, \frac{1}{5})$ [allow $Po(5)$]. May be written in full or implied by a correct CR |
| $P(X \geq 9) = 0.0468$, $P(X \leq 1) = 0.0274$ | A1 | Both awrt 0.0468 and awrt 0.0274 seen |
| $X = [0 \leq]\ X \leq 1$ | A1 | $X \leq 1$ or $X < 2$ or $0 \leq X \leq 1$ or $[0,1]$ or $0,1$ or equivalent statements. $X \leq c$ and $c = 1$ |
| $9 \leq X\ [\leq 25]$ | A1d | Dependent on seeing a probability from $B(25, 0.2)$ and $X \geq 9$ or $X > 8$ or $9 \leq X \leq 25$ or list $9,10,...,25$ or $[9,25]$ or equivalent. $X \geq c$ and $c = 9$ |
**NB:** Final 2 A marks must be for statements with "$X$" only (or list) — not in probability statements.
**SC:** If a probability from $B(25, 0.2)$ is seen **and** they have both CR correct but written as probability statements **or** CR written as $1 \geq X \geq 9$, award A1 A0 for final 2 marks. **(4)**
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## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: p = 0.2$, $H_1: p < 0.2$ | B1 | Both hypotheses with $p$ or $\pi$, clear which is $H_0$ and which is $H_1$ |
| $P(X \leq 6) = 0.1034$ or $CR\ X \leq 5$ | M1 | Writing or using $B(50, 0.2)$ and writing or using $P(X \leq 6)$ or $P(X \geq 7)$ on its own. May be implied by a correct CR |
| | A1 | awrt 0.103. Allow $CR\ X \leq 5$ or $X < 6$. If not using CR, allow awrt 0.897 |
| Insufficient evidence to reject $H_0$. Accept $H_0$, not significant. 6 does not lie in the critical region. | M1d | Dependent on previous M. Correct statement (do not allow contradicting non-contextual statements). ft their Prob/CR compared with $0.05/6/(0.95$ if using $0.8979)$. Do not follow through their hypotheses |
| No evidence that increasing the batch size has **reduced** the **percentage** of broken **pots (oe)** or evidence that there is **no change** in the **percentage** of broken **pots (oe)** | A1cso | Conclusion must contain words **reduced/no change/not affect oe number/percentage/proportion/probability oe**, and **pots**. All previous marks must be awarded for this mark. Do **not** allow potters claim/belief is wrong/true. **NB** Correct contextual statement on its own scores M1A1 |
**(5)**
**(Total 9)**\begin{enumerate}
\item A potter believes that $20 \%$ of pots break whilst being fired in a kiln. Pots are fired in batches of 25 .\\
(a) Let $X$ denote the number of broken pots in a batch. A batch is selected at random. Using a 10\% significance level, find the critical region for a two tailed test of the potter's belief. You should state the probability in each tail of your critical region.
\end{enumerate}
The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50 . He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.\\
(b) Test, at the $5 \%$ level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly.
\hfill \mbox{\textit{Edexcel S2 2017 Q1 [9]}}