## Question 12:

$H_0: p = 0.1$, $H_1: p \neq 0.1$ | B1 | AO 1.1 | allow equivalent in words; do not allow percentages; allow other variable only if correctly defined

$p$ is the probability that a (British) adult (selected at random) is a vegetarian | B1 | AO 2.5 | or $p$ is the proportion of adults that are vegetarian; **B1B1** if other symbol instead of $p$ used if correctly defined

$P(X \geq k)$ found using $B(112, 0.1)$, where $k = 18, 19$ or $20$ | M1* | AO 3.3 | may be implied by $P(X \geq 18) = 0.0295 - 0.030$ or $P(X \geq 19) = 0.015 - 0.015331$ or $P(X \geq 20) = 0.0075 - 0.00754$; **NB M0** for $P(X=19) = 0.00779$

$[P(X \geq 19) =] 0.015 - 0.015331$ | A1 | AO 1.1 | or $[P(X \leq 18) =] 0.984669 - 0.985$ or $0.98$

their $0.015$ correctly compared with $0.025$ **or** their $0.985$ correctly compared with $0.975$ | M1dep* | AO 3.4 |

do not accept $H_0$ **or** reject $H_0$ **or** accept $H_1$ **or** significant | A1FT | AO 1.1 | **A0** if their $0.015 > 0.025$ or their $0.985 < 0.975$

**sufficient evidence** at the 5% level to **suggest** that the **probability** that an adult is vegetarian **is not** $0.10$ **oe** | A1 | AO 3.5a | dependent on award of all other marks apart from second B1; do not allow e.g. conclude/prove/indicate or other assertive statement instead of suggest

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*Alternatively, using critical region:*

critical region is $X \leq k \cup X \geq l$, $X \geq k$ found from calculation of probability; allow $k = 4$ or $5$, $l = 18, 19$ or $20$ | M1* | | allow calculation of upper tail only for M1

[critical region is $X] \geq 19\ \cup\ [X] \leq 4$ | A1 | | from $P(X \geq 19) = 0.015 - 0.015331$ and $P(X \leq 4) = 0.010$; must see both tails for A1

$19$ correctly compared with their critical value | M1dep* | AO 3.4 |

do not accept $H_0$ **or** reject $H_0$ **or** accept $H_1$ **or** significant | A1FT | AO 1.1 | **A0** if $19 <$ their critical value

**sufficient evidence** at the 5% level to **suggest** that the **probability** that an adult is vegetarian **is not** $0.10$ **oe** | A1 | AO 3.5a |

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*Alternatively, using Normal approximation:*

$P(X \geq 18.5)$ or $P(X \geq 19.5)$ found using $N(11.2, 10.08)$ | M1* | | **NB M0** for $P(X=19) = 0.006145$

**or** $P(X \geq 19.5) = 0.00447(1)$

$[P(X \geq 18.5) =] 0.0107 - 0.011$ | A1 |

their $0.0107$ correctly compared with $0.025$ **or** their $0.989$ correctly compared with $0.975$ | M1dep* |

do not accept $H_0$ **or** reject $H_0$ **or** accept $H_1$ **or** significant | A1FT | | **A0** if their $0.0107 > 0.025$ or their $0.989 < 0.975$

**sufficient evidence** at the 5% level to **suggest** that the **probability** that an adult is vegetarian **is not** $0.10$ **oe** | A1 | AO 3.5a |

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