## Question 3:

### Part (i)(A)

$X \sim B(20, 0.25)$

$P(\text{4 smokers}) = \binom{20}{4} \times 0.25^4 \times 0.75^{16} = 0.1897$ | M1, M1, A1 [3] | M1 for $0.25^4 \times 0.75^{16}$; M1 for $\binom{20}{4} \times p^4 \times q^{16}$ with $p+q=1$; allow 0.19 or better. OR from tables: $0.4148 - 0.2252 = 0.1896$ | M2, A1 | For $0.4148 - 0.2252$; 0.189 gets A0

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### Part (i)(B)

$P(3 \leq X \leq 6) = 0.7858 - 0.0913 = 0.6945$ | M1, M1, A1 [3] | M1 for $P(X \leq 6) = 0.7858$; M1 for their $0.7858 - 0.0913$; accept 0.69 or better. Alternative: $P(X=3)+P(X=4)+P(X=5)+P(X=6) = 0.1339+0.1897+0.2023+0.1686 = 0.6945$; M1 for three correct terms. Also $P(X \geq 3) - P(X > 6) = 0.9087 - 0.2142 = 0.6945$ gets M1 M1 A1

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### Part (i)(C)

$E(X) = np = 20 \times 0.25 = 5$ | B1 [1] | CAO

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### Part (ii)(A)

Let $p$ = probability that a randomly selected student is a smoker

$H_0: p = 0.25$

$H_1: p < 0.25$ | B1, B1, B1 [3] | B1 for definition of $p$ in context; B1 for $H_0$; B1 for $H_1$. Allow complementary probabilities. Definition must include word probability/chance/proportion/percentage (NOT possibility). Do not allow $H_1: p \leq 0.25$. Allow $p=25\%$, allow $\theta$ or $\pi$ and $\rho$ but not $x$ unless defined.

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### Part (ii)(B)

$H_1$ has this form as the programme aims to reduce the **proportion** of smokers. | E1 [1] | Allow 'number'; allow 'aims for a reduction' or similar. E0 if $H_1$ upper tail or two tailed.

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### Part (iii)

$P(X \leq 1) = 0.0243 < 5\%$

$P(X \leq 2) = 0.0913 > 5\%$

So critical region is $\{0, 1\}$ | B1, B1, M1, A1 [4] | B1 for $P(X \leq 1) = 0.0243$; B1 for $P(X \leq 2) = 0.0913$; M1 for at least one comparison with 5%; A1 CAO for critical region, dep on M1 and at least one B1. NB use of point probabilities gets B0M0A0. Allow any form of statement of CR e.g. $X \leq 1$, $X < 2$, annotated number line etc.

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### Part (iv)

3 does not lie in the critical region, so not significant.

So there is not enough evidence to reject the null hypothesis and we conclude that there is not enough evidence to suggest that the percentage of smokers has decreased. | E1dep, E1dep [2] | E1dep for 3 not in CR or for not significant or reject $H_1$; E1dep for conclusion in context. Condone omission of 'not enough evidence' in this case. Dep on correct CR (correctly obtained). E0E0 for $P(X=3)$ not in CR. Alternative: $P(X \leq 3) = 0.2252 > 5\%$ so not sig etc. gets E2 for complete method but E0 otherwise.