## Question 3:

### Part (i)(A)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $X \sim B(10, 0.35)$ | | |
| $P(5 \text{ accessing internet}) = \binom{10}{5} \times 0.35^5 \times 0.65^5$ | M1 | or $0.35^5 \times 0.65^5$; with $p + q = 1$ |
| | M1 | For $\binom{10}{5} \times p^5 \times q^5$; also for $252 \times 0.0006094$ |
| $= 0.1536$ | A1 [3] | cao; allow 0.15 or better; NB 0.153 gets A0 |
| **OR** from tables $= 0.9051 - 0.7515 = 0.1536$ | M2, A1 | For $0.9051 - 0.7515$; cao |

### Part (i)(B)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 5) = 1 - P(X \leq 4)$ | M1 | For 0.7515 |
| $= 1 - 0.7515$ | | |
| $= 0.2485$ | A1 [2] | cao; accept 0.25 or better – allow 0.248 or 0.249; calculation of individual probabilities gets B2 if fully correct 0.25 or better, otherwise B0 |

### Part (i)(C)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = np = 10 \times 0.35$ | M1 | For $10 \times 0.35$ |
| $= 3.5$ | A1 [2] | cao; if any indication of rounding to 3 or 4 allow M1A0 |

### Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $X \sim B(20, 0.35)$ | | |
| Let $p$ = probability of a customer using the internet (for population) | B1 | For definition of $p$ in context; minimum needed is $p$ = probability of using internet; allow $p = P(\text{using internet})$; definition must include word probability (or chance or proportion or percentage or likelihood but NOT possibility); do NOT allow '$p$ = the probability of using internet is different' |
| $H_0: p = 0.35$ | B1 | Allow $p = 35\%$; allow only $p$ or $\theta$ or $\pi$ or $\rho$; allow any single symbol if defined; allow $H_0 = p = 0.35$; allow $p = \frac{7}{20}$ or $p = \frac{35}{100}$; do not allow $H_0: P(X=x) = 0.35$; do not allow $H_0$ and $H_1$ reversed |

## Question 3 (ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_1: p \neq 0.35$ | B1 | For $H_1$. Allow '$p < 0.35$ or $p > 0.35$' in place of $p \neq 0.35$. Do not allow if $H_1$ wrong. |
| $H_1$ has this form because the test is to investigate whether the proportion is different (rather than lower or higher). | E1 | |
| $P(X \geq 10)$ | B1 | For notation $P(X \geq 10)$ or $P(X > 9)$ or $1 - P(X \leq 9)$ (as long as no incorrect notation). This mark may be implied by 0.1218. No further marks if point probs used. $P(X=10) = 0.0686$ (do not even give notation mark for correct notation). DO NOT FT wrong $H_1$, but see extra notes. |
| $= 1 - 0.8782 = 0.1218$ | B1* | For 0.1218. Allow 0.12. Or for $1 - 0.8782$. Independent of previous mark. |
| $> 2.5$ | M1* dep | For comparison with 2.5% |
| So not significant. Conclude that there is not enough evidence to indicate that the probability is different. (Must state 'probability', not just 'p') | A1* E1* dep on A1 | Allow 'accept $H_0$' or 'reject $H_1$'. Must include 'sufficient evidence' or something similar such as 'to suggest that' ie an element of doubt either in the A or E mark. |

**Alternative Method (Critical Region):**

| Answer | Marks | Guidance |
|--------|-------|----------|
| LOWER TAIL: $P(X \leq 2) = 0.0121 < 2.5\%$; $P(X \leq 3) = 0.0444 > 2.5\%$ | B1 | For either probability. Do not insist on correct notation as candidates have to work out two probabilities for full marks. If only upper tail of CR given (or only upper tail justified), allow max 4/5 for final 5 marks. |
| UPPER TAIL: $P(X \geq 11) = 1 - P(X \leq 10) = 1 - 0.9468 = 0.0532 > 2.5\%$; $P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.9804 = 0.0196 < 2.5\%$ | B1 | For either probability |
| So critical region is $\{0,1,2,12,13,14,15,16,17,18,19,20\}$ | M1* dep | cao dep on at least one correct comparison with 2.5%. No marks if CR not justified. Condone $\{0,1,2,12,\ldots 20\}$, $X \leq 2$, $X \geq 12$, oe but not $P(X \leq 2)$ etc |
| So not significant. Conclude that there is not enough evidence to indicate that the probability is different. | A1* E1* dep on A1 | NB If CR found correctly then $P(X=10)$ subsequently found but candidate says '10 not in CR' then allow up to all last five marks. If do not say '10 not in CR' allow none of last five marks. |

**[9 marks total]**

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## Question 3 (iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.0022 < 2.5\%$, so reject $H_0$. Significant. | B1 | For either reject $H_0$ or significant, dep on correct comparison |
| Conclude that there is enough evidence to indicate that the probability is different. | E1* dep | Dep on good attempt at correct hypotheses in part (ii). If they have $H_1: p > 0.35$, allow SC1 if all correct including comparison with 5%. |

**[2 marks total]**