| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward application of binomial hypothesis testing with standard calculations. Part (i) involves routine binomial probability computations, part (ii) is a textbook one-tailed test requiring comparison of P(X≥19) to 1%, and part (iii) simply reinterprets the same calculation at a different significance level. All steps are procedural 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 probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X \geq 19) = 0.0392 + 0.0069\) | B1 | For notation \(P(X \geq 19)\) or \(P(X > 18)\) or \(1 - P(X \leq 18)\) or \(1 - P(X < 19)\). Notation \(P(X = 19)\) scores B0. If correct \(P(X \geq 19)\) then give B1 and ignore any further incorrect notation. |
| \(= 0.0461\) | B1* | CAO. Allow 0.0462 |
| \(0.0461 > 1\%\), so not significant | M1* dep | For comparison with 1%. Dep on sensible attempt at \(P(X \geq 19)\) |
| Conclude that there is not enough evidence to suggest that the new drug is more effective than the old one | A1 | Allow 'accept \(H_0\)' or 'reject \(H_1\)'. Must include 'insufficient evidence' or similar such as 'to suggest that'. Must be in context. Do NOT allow 'sufficient evidence to suggest proportion cured is 0.78' or similar |
| E1 | 99% method: \(P(X \leq 18) = 0.9539\) B1B1* CAO; \(0.9539 < 99\%\) M1* then as per scheme |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X \geq 19) = 0.0461 > 1\%\) | B1 | For either probability. Do not insist on correct notation as candidates have to work out two probabilities for full marks. No further marks if point probabilities used |
| \(P(X \geq 20) = 0.0069 < 1\%\) | M1 | For at least one comparison with 1%. Allow comparison in form of statement 'critical region at 1% level is …' |
| So critical region is \(\{20\}\) | B1* | CAO dep on the two correct probabilities. No marks if CR not justified. Condone \(X \geq 20\), \(X = 20\), oe but not \(P(X \geq 20)\) etc |
| (19 not in CR so) not significant | A1* dep | Dep on correct CR. Allow 'accept \(H_0\)' or 'reject \(H_1\)' |
| Conclude that there is not enough evidence to suggest that the new drug is more effective than the old one | E1* dep | Ignore any work on lower critical region |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| With a 5% significance level rather than a 1% level, the null hypothesis would have been rejected. OR: 'there would be enough evidence to suggest that the new drug is more effective than the old one.' | B1* | oe. FT their probability from (ii) but NO marks if point probabilities used. There must be a sensible attempt to use \(P(X = 19) + P(X = 20)\) or must have correct CR. |
| This is because \(0.0461 < 5\%\) | B1* dep | oe. Dep on correct answer of 0.0461 compared with 5% or 0.9539 compared with 95% or correct CR. |
# Question 1:
## Part (i) [continued]
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 19) = 0.0392 + 0.0069$ | B1 | For **notation** $P(X \geq 19)$ or $P(X > 18)$ or $1 - P(X \leq 18)$ or $1 - P(X < 19)$. Notation $P(X = 19)$ scores B0. If correct $P(X \geq 19)$ then give B1 and ignore any further incorrect notation. |
| $= 0.0461$ | B1* | CAO. Allow 0.0462 |
| $0.0461 > 1\%$, so not significant | M1* dep | For comparison with 1%. Dep on sensible attempt at $P(X \geq 19)$ |
| Conclude that there is not enough evidence to suggest that the new drug is more effective than the old one | A1 | Allow 'accept $H_0$' or 'reject $H_1$'. Must include 'insufficient evidence' or similar such as 'to suggest that'. Must be in context. Do NOT allow 'sufficient evidence to suggest proportion cured is 0.78' or similar |
| | E1 | 99% method: $P(X \leq 18) = 0.9539$ B1B1* CAO; $0.9539 < 99\%$ M1* then as per scheme |
**Alternative Method for Final 5 Marks:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 19) = 0.0461 > 1\%$ | B1 | For either probability. Do not insist on correct notation as candidates have to work out two probabilities for full marks. No further marks if point probabilities used |
| $P(X \geq 20) = 0.0069 < 1\%$ | M1 | For at least one comparison with 1%. Allow comparison in form of statement 'critical region at 1% level is …' |
| So critical region is $\{20\}$ | B1* | CAO dep on the two correct probabilities. No marks if CR not justified. Condone $X \geq 20$, $X = 20$, oe but not $P(X \geq 20)$ etc |
| (19 not in CR so) not significant | A1* dep | Dep on correct CR. Allow 'accept $H_0$' or 'reject $H_1$' |
| Conclude that there is not enough evidence to suggest that the new drug is more effective than the old one | E1* dep | Ignore any work on lower critical region |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| With a 5% significance level rather than a 1% level, the null hypothesis would have been rejected. OR: 'there would be enough evidence to suggest that the new drug is more effective than the old one.' | B1* | oe. FT their probability from (ii) but NO marks if point probabilities used. There must be a sensible attempt to use $P(X = 19) + P(X = 20)$ or must have correct CR. |
| This is because $0.0461 < 5\%$ | B1* dep | oe. Dep on correct answer of 0.0461 compared with 5% or 0.9539 compared with 95% or correct CR. |
---1 A drug for treating a particular minor illness cures, on average, $78 \%$ of patients. Twenty people with this minor illness are selected at random and treated with the drug.
\begin{enumerate}[label=(\roman*)]
\item (A) Find the probability that exactly 19 patients are cured.\\
(B) Find the probability that at most 18 patients are cured.\\
(C) Find the expected number of patients who are cured.
\item A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the $1 \%$ significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
\item If the researchers had chosen to carry out the hypothesis test at the $5 \%$ significance level, what would the result have been? Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 Q1 [17]}}