## Question 2:

### Part (i)(B)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Exactly 20 cured}) = \binom{20}{20} \times 0.78^{20} \times 0.22^{0} = 0.0069$ | M1 | For $0.78^{20}$ oe |
| $P(\text{At most 18 cured}) = 1 - (0.0069 + 0.0392)$ | M1 | For $P(19) + P(20)$; not necessarily correct, but both correct attempts at binomial, including coefficient in (i) and no extra terms (such as $P(X=18)$); condone use of $p = 0.8$ |
| $= 0.954$ (0.95385) | A1 [3] | CAO; allow 0.95 with working |

### Part (i)(C)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = np = 20 \times 0.78 = 15.6$ | B1 [1] | CAO; do not allow final answer of 15 or 16 even if correct 15.6 given earlier |

### Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $X \sim B(20, 0.78)$ | | |
| Let $p$ = probability of a patient being cured (for population) | B1 | For definition of $p$ in context |
| $H_0: p = 0.78$ | B1 | For $H_0$ |
| $H_1: p > 0.78$ | B1 | For $H_1$; no further marks if point probabilities used |

### Part (ii) 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 |
| $= 0.0461$ | B1* | CAO; allow 0.0462 |
| $0.0461 > 1\%$ | M1* dep | For comparison with 1% |
| So not significant. | A1 | 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 [8] | Must include 'insufficient evidence' or similar element of doubt; in context; do not allow 'sufficient evidence to suggest proportion cured is 0.78' |

**99% method:** $P(X \leq 18) = 0.9539$; B1B1* CAO; $0.9539 < 99\%$ M1* then as per scheme

**Alternative Method:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 20) = 0.0069 < 1\%$ | M1 | For at least one comparison with 1% |
| So critical region is $\{20\}$ | B1* | CAO dep on the two correct probabilities |
| (19 not in CR so) not significant | A1* dep | Dep on correct CR |
| 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; must have sensible attempt to use $P(X=19) + P(X=20)$ or must have correct CR |
| This is because $0.0461 < 5\%$ | B1* dep [2] | oe; dep on correct answer of 0.0461 compared with 5% or 0.9539 compared with 95% or correct CR |

---