| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find sample size for test |
| Difficulty | Standard +0.3 This is a straightforward S1 hypothesis testing question with standard binomial calculations. Part (i) involves routine probability calculations, part (ii) requires solving P(X≥1)≥0.8 by trial, and part (iii) is a textbook one-tailed binomial test. All techniques are standard with no novel insight required, making it slightly easier than average. |
| 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 | Marks | Guidance |
| \(X \sim B(20, 0.1)\) | ||
| (A) \(P(X=1) = \binom{20}{1} \times 0.1 \times 0.9^{19} = 0.2702\) | M1 | \(0.1 \times 0.9^{19}\) |
| M1 | \(\binom{20}{1} \times pq^{19}\) | |
| A1 CAO | OR: M2 for \(0.3917 - 0.1216\), A1 CAO. Total: 3 marks | |
| (B) \(P(X \geq 1) = 1 - 0.1216 = 0.8784\) | M1 | \(P(X=0)\) provided \(P(X \geq 1) = 1 - P(X \leq 1)\) not seen |
| M1 | \(1 - P(X=0)\) | |
| A1 CAO | Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: \(1 - 0.9^n \geq 0.8\), so \(0.9^n \leq 0.2\), minimum \(n = 16\) | M1 | for \(0.9^n\) |
| M1 | for inequality | |
| A1 CAO | ||
| OR (trial and improvement): trial with \(0.9^{15}\) or \(0.9^{16}\) or \(0.9^{17}\); \(1 - 0.9^{15} = 0.7941 < 0.8\) and \(1 - 0.9^{16} = 0.8147 > 0.8\); minimum \(n = 16\) | M1, M1 | |
| A1 CAO | NOTE: \(n=16\) unsupported scores SC1 only. Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (A) Let \(p\) = probability of a randomly selected rock containing a fossil (for population) | B1 | for definition of \(p\) |
| \(H_0: p = 0.1\) | B1 | for \(H_0\) |
| \(H_1: p < 0.1\) | B1 | for \(H_1\). Total: 3 marks |
| (B) Let \(X \sim B(30, 0.1)\); \(P(X \leq 0) = 0.0424 < 5\%\); \(P(X \leq 1) = 0.0424 + 0.1413 = 0.1837 > 5\%\) | M1 | attempt to find \(P(X \leq 0)\) or \(P(X \leq 1)\) using binomial |
| So critical region consists only of \(\{0\}\) | M1 | for both attempted |
| M1 | for comparison of either with 5% | |
| A1 | for critical region dep on both comparisons (NB Answer given). Total: 4 marks | |
| (C) 2 does not lie in the critical region, so there is insufficient evidence to reject the null hypothesis and we conclude that it seems that 10% of rocks in this area contain fossils. | M1 | for comparison |
| A1 | for conclusion in context. Total: 2 marks |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim B(20, 0.1)$ | | |
| (A) $P(X=1) = \binom{20}{1} \times 0.1 \times 0.9^{19} = 0.2702$ | M1 | $0.1 \times 0.9^{19}$ |
| | M1 | $\binom{20}{1} \times pq^{19}$ |
| | A1 CAO | OR: M2 for $0.3917 - 0.1216$, A1 CAO. Total: 3 marks |
| (B) $P(X \geq 1) = 1 - 0.1216 = 0.8784$ | M1 | $P(X=0)$ provided $P(X \geq 1) = 1 - P(X \leq 1)$ not seen |
| | M1 | $1 - P(X=0)$ |
| | A1 CAO | Total: 3 marks |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| EITHER: $1 - 0.9^n \geq 0.8$, so $0.9^n \leq 0.2$, minimum $n = 16$ | M1 | for $0.9^n$ |
| | M1 | for inequality |
| | A1 CAO | |
| OR (trial and improvement): trial with $0.9^{15}$ or $0.9^{16}$ or $0.9^{17}$; $1 - 0.9^{15} = 0.7941 < 0.8$ and $1 - 0.9^{16} = 0.8147 > 0.8$; minimum $n = 16$ | M1, M1 | |
| | A1 CAO | NOTE: $n=16$ unsupported scores SC1 only. Total: 3 marks |
### Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (A) Let $p$ = probability of a randomly selected rock containing a fossil (for population) | B1 | for definition of $p$ |
| $H_0: p = 0.1$ | B1 | for $H_0$ |
| $H_1: p < 0.1$ | B1 | for $H_1$. Total: 3 marks |
| (B) Let $X \sim B(30, 0.1)$; $P(X \leq 0) = 0.0424 < 5\%$; $P(X \leq 1) = 0.0424 + 0.1413 = 0.1837 > 5\%$ | M1 | attempt to find $P(X \leq 0)$ or $P(X \leq 1)$ using binomial |
| So critical region consists only of $\{0\}$ | M1 | for both attempted |
| | M1 | for comparison of either with 5% |
| | A1 | for critical region dep on both comparisons **(NB Answer given)**. Total: 4 marks |
| (C) 2 does not lie in the critical region, so there is insufficient evidence to reject the null hypothesis and we conclude that it seems that 10% of rocks in this area contain fossils. | M1 | for comparison |
| | A1 | for conclusion **in context**. Total: 2 marks |4 A geologist splits rocks to look for fossils. On average 10\% of the rocks selected from a particular area do in fact contain fossils.
The geologist selects a random sample of 20 rocks from this area.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that\\
(A) exactly one of the rocks contains fossils,\\
(B) at least one of the rocks contains fossils.
\item A random sample of $n$ rocks is selected from this area. The geologist wants to have a probability of 0.8 or greater of finding fossils in at least one of the $n$ rocks. Find the least possible value of $n$.
\item The geologist explores a new area in which it is claimed that less than $10 \%$ of rocks contain fossils. In order to investigate the claim, a random sample of 30 rocks from this area is selected, and the number which contain fossils is recorded. A hypothesis test is carried out at the 5\% level.\\
(A) Write down suitable hypotheses for the test.\\
(B) Show that the critical region consists only of the value 0 .\\
(C) In fact, 2 of the 30 rocks in the sample contain fossils. Complete the test, stating your conclusions clearly.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 Q4 [15]}}