# Question 7(i)(A):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim B(20, 0.78)$ | | |
| $P(\text{Exactly 19 cured})=\binom{20}{19}\times 0.78^{19}\times 0.22^1$ | M1 | For $0.78^{19}\times 0.22^1$ |
| | M1 | For $\binom{20}{19}\times p^{19}\times q^1$ with $p+q=1$ |
| $=0.0392$ (0.039197) | A1 | CAO; allow 0.039 or better; condone 0.03919 but not 0.0391 |
| **[3]** | | |

---

# Question 7(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 $0.78^{20}$; allow M2 for 0.9488 for linear interpolation |
| $P(\text{At most 18 cured})=1-(0.0069+0.0392)$ | M1 | For $P(19)+P(20)$; zero for use of $p=0.8$ |
| $=0.954$ (0.95385) | A1 | CAO; allow 0.95 with working |
| **[3]** | | |

---

# Question 7(i)(C):

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

---

# Question 7(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 |
| $P(X\geq 19)=0.0392+0.0069$ | B1 | For **NOTATION** $P(X\geq 19)$ or $P(X>18)$ or $1-P(X\leq 18)$ or $1-P(X<19)$ |
| $=0.0461$ | 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 | Must include 'insufficient evidence' or similar element of doubt; must be in context; do NOT allow 'sufficient evidence to suggest proportion cured is 0.78' |
| 99% method: $P(X\leq 18)=0.9539$; $0.9539<99\%$ then as per scheme | | |
| **[8]** | | |

---

# Question 7(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| With a 5% significance level rather than 1%, 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* | FT their probability from (ii) but NO marks if point probabilities used; sensible attempt to use $P(X=19)+P(X=20)$ or must have correct CR |
| This is because $0.0461<5\%$ | B1* dep | Dep on correct answer of 0.0461 compared with 5% or 0.9539 compared with 95% or correct CR |
| **[2]** | | |

Alternative method: $P(X\geq 20)=0.0069<1\%$ | M1 | For at least one comparison with 1% |
| So critical region is $\{20\}$ | B1* | CAO dep on two correct probabilities |
| (19 not in CR so) not significant | A1* dep | Dep on correct CR |
| Conclude not enough evidence to suggest new drug more effective | E1* dep | Ignore any work on lower critical region |

---