| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find sample size for test |
| Difficulty | Standard +0.8 Part (a) requires understanding critical region construction for binomial tests and working backwards to find minimum sample size, which is non-routine. Part (b) is a standard two-tailed hypothesis test application. The combination of reverse-engineering sample size plus standard testing makes this moderately challenging but accessible to well-prepared students. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.94^n < k\) or \(0.06^n < k\) seen | M1 (3.4) | Allow \(=\) instead of \(<\) |
| \(k = 0.025\) used in inequality as above | B1 (1.1) | |
| 60 | A1 (2.2a) | NB 59.617.. or 1.311…to 1 or more dp if unsupported implies M1B1; 60 unsupported or from trial and improvement scores 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p = 0.06\) (allow equivalent in words) | B1 | |
| \(H_1: p \neq 0.06\) | ||
| \(p\) is the probability that a jaguar chosen at random is a black panther / has black coat | B1 | or \(p\) is the proportion of jaguars that are black panthers / have a black coat |
| Use of \(B(83, 0.06)\) to obtain \(P(X \leq K)\) oe | M1* | not \(P(X=K)\); NB \(P(X \leq 10) = .98927\ldots\) |
| \(\text{cdfBinomial}(83, 0.06, 9) = 0.973\) to \(0.97321\ldots\) or \(1 - \text{cdfBinomial}(83, 0.06, 9) = 0.02679\) to \(0.027\) | A1 | or critical region is \(X \geq 11\) (ignore lower tail); for comparison of their \(P(X>K)\) with \(0.025\) or their \(P(X \leq K)\) with \(0.975\) |
| \(1 -\) their \(P(X \leq K)\) compared with \(0.025\) or their \(P(X \leq K)\) compared with \(0.975\) oe | M1dep* | eg 10 compared with their critical region oe; their \(P(X \leq K)\) with \(0.975\) or stating whether 10 is in their critical region |
| Result is not significant or do not reject \(H_0\) or reject \(H_1\) | A1 | must have correct probability or correct critical region for last two A marks; allow accept \(H_0\) |
| There is insufficient evidence at the 5% level to suggest that the probability that a jaguar selected at random from this population is a black panther is not 0.06 | A1 | do not allow eg conclude / prove / indicate or other assertive statement instead of suggest |
## Question 12(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.94^n < k$ or $0.06^n < k$ seen | M1 (3.4) | Allow $=$ instead of $<$ |
| $k = 0.025$ used in inequality as above | B1 (1.1) | |
| 60 | A1 (2.2a) | NB 59.617.. or 1.311…to 1 or more dp if unsupported implies M1B1; 60 unsupported or from trial and improvement scores 3 |
## Question 12(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = 0.06$ (allow equivalent in words) | B1 | |
| $H_1: p \neq 0.06$ | | |
| $p$ is the probability that a **jaguar** chosen at random is a **black** panther / has **black** coat | B1 | or $p$ is the proportion of **jaguars** that are **black** panthers / have a **black** coat |
| Use of $B(83, 0.06)$ to obtain $P(X \leq K)$ oe | M1* | not $P(X=K)$; NB $P(X \leq 10) = .98927\ldots$ |
| $\text{cdfBinomial}(83, 0.06, 9) = 0.973$ to $0.97321\ldots$ or $1 - \text{cdfBinomial}(83, 0.06, 9) = 0.02679$ to $0.027$ | A1 | or critical region is $X \geq 11$ (ignore lower tail); for comparison of their $P(X>K)$ with $0.025$ or their $P(X \leq K)$ with $0.975$ |
| $1 -$ their $P(X \leq K)$ compared with $0.025$ or their $P(X \leq K)$ compared with $0.975$ oe | M1dep* | eg 10 compared with their critical region oe; their $P(X \leq K)$ with $0.975$ or stating whether 10 is in their critical region |
| Result is not significant or do not reject $H_0$ or reject $H_1$ | A1 | must have correct probability or correct critical region for last two A marks; allow accept $H_0$ |
| There is insufficient evidence at the 5% level to **suggest** that the probability that a **jaguar** selected at random from this population is a **black panther** is not 0.06 | A1 | do not allow eg conclude / prove / indicate or other assertive statement instead of suggest |
---12 The jaguar is a species of big cat native to South America. Records show that 6\% of jaguars are born with black coats. Jaguars with black coats are known as black panthers. Due to deforestation a population of jaguars has become isolated in part of the Amazon basin. Researchers believe that the percentage of black panthers may not be $6 \%$ in this population.
\begin{enumerate}[label=(\alph*)]
\item Find the minimum sample size needed to conduct a two-tailed test to determine whether there is any evidence at the $5 \%$ level to suggest that the percentage of black panthers is not $6 \%$.
A research team identifies 70 possible sites for monitoring the jaguars remotely. 30 of these sites are randomly selected and cameras are installed. 83 different jaguars are filmed during the evidence gathering period. The team finds that 10 of the jaguars are black panthers.
\item Conduct a hypothesis test to determine whether the information gathered by the research team provides any evidence at the $5 \%$ level to suggest that the percentage of black panthers in this population is not $6 \%$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2019 Q12 [10]}}