| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| 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 covering standard binomial probability calculations and a one-tailed test. Parts (i) and (ii) are routine binomial calculations, while part (iii) follows a standard hypothesis test template with clearly defined steps. The critical region calculation requires some care but is a standard technique. Slightly above average difficulty due to the multi-part nature and need to work with cumulative probabilities, but no novel insight required. |
| 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 |
|---|---|
| \(X \sim B(20, 0.1)\) | M1 |
| \(P(X=1) = \binom{20}{1}(0.1)^1(0.9)^{19} = 20 \times 0.1 \times 0.9^{19}\) | M1 |
| \(= 0.2702\) | A1 |
| Answer | Marks |
|---|---|
| \(P(X \geq 1) = 1 - P(X=0) = 1 - 0.9^{20}\) | M1 A1 |
| \(= 1 - 0.1216 = 0.8784\) | A1 |
| Answer | Marks |
|---|---|
| Need \(P(X \geq 1) \geq 0.8\), i.e. \(1 - 0.9^n \geq 0.8\) | M1 |
| \(0.9^n \leq 0.2\) | M1 |
| \(n \geq \frac{\ln 0.2}{\ln 0.9} = 15.27...\) so \(n = 16\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.1\) | B1 | |
| \(H_1: p < 0.1\) | B1 | One-tailed |
| Where \(p\) is probability a rock contains fossils | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=0) = 0.9^{30} = 0.0424 < 0.05\) | M1 A1 | |
| \(P(X \leq 1) = 0.9^{30} + 30(0.1)(0.9)^{29} = 0.0424 + 0.1413 = 0.1837 > 0.05\) | M1 A1 | |
| Critical region is \(\{0\}\) only | Shown |
| Answer | Marks |
|---|---|
| \(x = 2\), which is not in critical region \(\{0\}\) | M1 |
| Do not reject \(H_0\); insufficient evidence that proportion of rocks with fossils is less than 10% | A1 |
# Question 7:
## Part (i)(A)
| $X \sim B(20, 0.1)$ | M1 | |
| $P(X=1) = \binom{20}{1}(0.1)^1(0.9)^{19} = 20 \times 0.1 \times 0.9^{19}$ | M1 | |
| $= 0.2702$ | A1 | |
## Part (i)(B)
| $P(X \geq 1) = 1 - P(X=0) = 1 - 0.9^{20}$ | M1 A1 | |
| $= 1 - 0.1216 = 0.8784$ | A1 | |
## Part (ii)
| Need $P(X \geq 1) \geq 0.8$, i.e. $1 - 0.9^n \geq 0.8$ | M1 | |
| $0.9^n \leq 0.2$ | M1 | |
| $n \geq \frac{\ln 0.2}{\ln 0.9} = 15.27...$ so $n = 16$ | A1 | |
## Part (iii)(A)
| $H_0: p = 0.1$ | B1 | |
| $H_1: p < 0.1$ | B1 | One-tailed |
| Where $p$ is probability a rock contains fossils | B1 | |
## Part (iii)(B)
| $P(X=0) = 0.9^{30} = 0.0424 < 0.05$ | M1 A1 | |
| $P(X \leq 1) = 0.9^{30} + 30(0.1)(0.9)^{29} = 0.0424 + 0.1413 = 0.1837 > 0.05$ | M1 A1 | |
| Critical region is $\{0\}$ only | | Shown |
## Part (iii)(C)
| $x = 2$, which is not in critical region $\{0\}$ | M1 | |
| Do not reject $H_0$; insufficient evidence that proportion of rocks with fossils is less than 10% | A1 | |7 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 2006 Q7 [18]}}